Calculus Examples

$\frac{d y}{d x} = \frac{x^{3} - 21 x}{5 + 4 x - x^{2}} x^{3}$

Step 1

Rewrite the equation.

$d y = \frac{x^{3} - 21 x}{5 + 4 x - x^{2}} x^{3} d x$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

$\int d y = \int \frac{x^{3} - 21 x}{5 + 4 x - x^{2}} x^{3} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int \frac{x^{3} - 21 x}{5 + 4 x - x^{2}} x^{3} d x$

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

Combine $\frac{x^{3} - 21 x}{5 + 4 x - x^{2}}$ and $x^{3}$ .

$y + C_{1} = \int \frac{(x^{3} - 21 x) x^{3}}{5 + 4 x - x^{2}} d x$

Step 2.3.2

Apply the distributive property.

$y + C_{1} = \int \frac{x^{3} x^{3} - 21 x \cdot x^{3}}{5 + 4 x - x^{2}} d x$

Step 2.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y + C_{1} = \int \frac{x^{3 + 3} - 21 x \cdot x^{3}}{5 + 4 x - x^{2}} d x$

Step 2.3.4

Add $3$ and $3$ .

$y + C_{1} = \int \frac{x^{6} - 21 x \cdot x^{3}}{5 + 4 x - x^{2}} d x$

Step 2.3.5

Raise $x$ to the power of $1$ .

$y + C_{1} = \int \frac{x^{6} - 21 (x^{1} x^{3})}{5 + 4 x - x^{2}} d x$

Step 2.3.6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y + C_{1} = \int \frac{x^{6} - 21 x^{1 + 3}}{5 + 4 x - x^{2}} d x$

Step 2.3.7

Add $1$ and $3$ .

$y + C_{1} = \int \frac{x^{6} - 21 x^{4}}{5 + 4 x - x^{2}} d x$

Step 2.3.8

Reorder $5$ and $4 x$ .

$y + C_{1} = \int \frac{x^{6} - 21 x^{4}}{4 x + 5 - x^{2}} d x$

Step 2.3.9

Move $5$ .

$y + C_{1} = \int \frac{x^{6} - 21 x^{4}}{4 x - x^{2} + 5} d x$

Step 2.3.10

Reorder $4 x$ and $- x^{2}$ .

$y + C_{1} = \int \frac{x^{6} - 21 x^{4}}{- x^{2} + 4 x + 5} d x$

Step 2.3.11

Divide $x^{6} - 21 x^{4}$ by $- x^{2} + 4 x + 5$ .

Tap for more steps...

Step 2.3.11.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x^{2}$

$4 x$

$5$

$x^{6}$

$0 x^{5}$

$21 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

Step 2.3.11.2

Divide the highest order term in the dividend $x^{6}$ by the highest order term in divisor $- x^{2}$ .

$x^{4}$

$x^{2}$

$4 x$

$5$

$x^{6}$

$0 x^{5}$

$21 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

Step 2.3.11.3

Multiply the new quotient term by the divisor.

$x^{4}$

$x^{2}$

$4 x$

$5$

$x^{6}$

$0 x^{5}$

$21 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{6}$

$4 x^{5}$

$5 x^{4}$

Step 2.3.11.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{6} - 4 x^{5} - 5 x^{4}$

$x^{4}$

$x^{2}$

$4 x$

$5$

$x^{6}$

$0 x^{5}$

$21 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{6}$

$4 x^{5}$

$5 x^{4}$

Step 2.3.11.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{4}$

$x^{2}$

$4 x$

$5$

$x^{6}$

$0 x^{5}$

$21 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{6}$

$4 x^{5}$

$5 x^{4}$

$4 x^{5}$

$16 x^{4}$

Step 2.3.11.6

Pull the next terms from the original dividend down into the current dividend.

$x^{4}$

$x^{2}$

$4 x$

$5$

$x^{6}$

$0 x^{5}$

$21 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{6}$

$4 x^{5}$

$5 x^{4}$

$4 x^{5}$

$16 x^{4}$

$0 x^{3}$

Step 2.3.11.7

Divide the highest order term in the dividend $4 x^{5}$ by the highest order term in divisor $- x^{2}$ .

$x^{4}$

$4 x^{3}$

$x^{2}$

$4 x$

$5$

$x^{6}$

$0 x^{5}$

$21 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{6}$

$4 x^{5}$

$5 x^{4}$

$4 x^{5}$

$16 x^{4}$

$0 x^{3}$

Step 2.3.11.8

Multiply the new quotient term by the divisor.

$x^{4}$

$4 x^{3}$

$x^{2}$

$4 x$

$5$

$x^{6}$

$0 x^{5}$

$21 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{6}$

$4 x^{5}$

$5 x^{4}$

$4 x^{5}$

$16 x^{4}$

$0 x^{3}$

$4 x^{5}$

$16 x^{4}$

$20 x^{3}$

Step 2.3.11.9

The expression needs to be subtracted from the dividend, so change all the signs in $4 x^{5} - 16 x^{4} - 20 x^{3}$

$x^{4}$

$4 x^{3}$

$x^{2}$

$4 x$

$5$

$x^{6}$

$0 x^{5}$

$21 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{6}$

$4 x^{5}$

$5 x^{4}$

$4 x^{5}$

$16 x^{4}$

$0 x^{3}$

$4 x^{5}$

$16 x^{4}$

$20 x^{3}$

Step 2.3.11.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{4}$

$4 x^{3}$

$x^{2}$

$4 x$

$5$

$x^{6}$

$0 x^{5}$

$21 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{6}$

$4 x^{5}$

$5 x^{4}$

$4 x^{5}$

$16 x^{4}$

$0 x^{3}$

$4 x^{5}$

$16 x^{4}$

$20 x^{3}$

Step 2.3.11.11

Pull the next term from the original dividend down into the current dividend.

$x^{4}$

$4 x^{3}$

$x^{2}$

$4 x$

$5$

$x^{6}$

$0 x^{5}$

$21 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{6}$

$4 x^{5}$

$5 x^{4}$

$4 x^{5}$

$16 x^{4}$

$0 x^{3}$

$4 x^{5}$

$16 x^{4}$

$20 x^{3}$

$0 x^{2}$

$0 x$

Step 2.3.11.12

Divide the highest order term in the dividend $20 x^{3}$ by the highest order term in divisor $- x^{2}$ .

$x^{4}$

$4 x^{3}$

$0 x^{2}$

$20 x$

$x^{2}$

$4 x$

$5$

$x^{6}$

$0 x^{5}$

$21 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{6}$

$4 x^{5}$

$5 x^{4}$

$4 x^{5}$

$16 x^{4}$

$0 x^{3}$

$4 x^{5}$

$16 x^{4}$

$20 x^{3}$

$0 x^{2}$

$0 x$

Step 2.3.11.13

Multiply the new quotient term by the divisor.

$x^{4}$

$4 x^{3}$

$0 x^{2}$

$20 x$

$x^{2}$

$4 x$

$5$

$x^{6}$

$0 x^{5}$

$21 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{6}$

$4 x^{5}$

$5 x^{4}$

$4 x^{5}$

$16 x^{4}$

$0 x^{3}$

$4 x^{5}$

$16 x^{4}$

$20 x^{3}$

$0 x^{2}$

$0 x$

$20 x^{3}$

$80 x^{2}$

$100 x$

Step 2.3.11.14

The expression needs to be subtracted from the dividend, so change all the signs in $20 x^{3} - 80 x^{2} - 100 x$

$x^{4}$

$4 x^{3}$

$0 x^{2}$

$20 x$

$x^{2}$

$4 x$

$5$

$x^{6}$

$0 x^{5}$

$21 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{6}$

$4 x^{5}$

$5 x^{4}$

$4 x^{5}$

$16 x^{4}$

$0 x^{3}$

$4 x^{5}$

$16 x^{4}$

$20 x^{3}$

$0 x^{2}$

$0 x$

$20 x^{3}$

$80 x^{2}$

$100 x$

Step 2.3.11.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{4}$

$4 x^{3}$

$0 x^{2}$

$20 x$

$x^{2}$

$4 x$

$5$

$x^{6}$

$0 x^{5}$

$21 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{6}$

$4 x^{5}$

$5 x^{4}$

$4 x^{5}$

$16 x^{4}$

$0 x^{3}$

$4 x^{5}$

$16 x^{4}$

$20 x^{3}$

$0 x^{2}$

$0 x$

$20 x^{3}$

$80 x^{2}$

$100 x$

$80 x^{2}$

$100 x$

Step 2.3.11.16

Pull the next terms from the original dividend down into the current dividend.

$x^{4}$

$4 x^{3}$

$0 x^{2}$

$20 x$

$x^{2}$

$4 x$

$5$

$x^{6}$

$0 x^{5}$

$21 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{6}$

$4 x^{5}$

$5 x^{4}$

$4 x^{5}$

$16 x^{4}$

$0 x^{3}$

$4 x^{5}$

$16 x^{4}$

$20 x^{3}$

$0 x^{2}$

$0 x$

$20 x^{3}$

$80 x^{2}$

$100 x$

$80 x^{2}$

$100 x$

$0$

Step 2.3.11.17

Divide the highest order term in the dividend $80 x^{2}$ by the highest order term in divisor $- x^{2}$ .

$x^{4}$

$4 x^{3}$

$0 x^{2}$

$20 x$

$80$

$x^{2}$

$4 x$

$5$

$x^{6}$

$0 x^{5}$

$21 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{6}$

$4 x^{5}$

$5 x^{4}$

$4 x^{5}$

$16 x^{4}$

$0 x^{3}$

$4 x^{5}$

$16 x^{4}$

$20 x^{3}$

$0 x^{2}$

$0 x$

$20 x^{3}$

$80 x^{2}$

$100 x$

$80 x^{2}$

$100 x$

$0$

Step 2.3.11.18

Multiply the new quotient term by the divisor.

$x^{4}$

$4 x^{3}$

$0 x^{2}$

$20 x$

$80$

$x^{2}$

$4 x$

$5$

$x^{6}$

$0 x^{5}$

$21 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{6}$

$4 x^{5}$

$5 x^{4}$

$4 x^{5}$

$16 x^{4}$

$0 x^{3}$

$4 x^{5}$

$16 x^{4}$

$20 x^{3}$

$0 x^{2}$

$0 x$

$20 x^{3}$

$80 x^{2}$

$100 x$

$80 x^{2}$

$100 x$

$0$

$80 x^{2}$

$320 x$

$400$

Step 2.3.11.19

The expression needs to be subtracted from the dividend, so change all the signs in $80 x^{2} - 320 x - 400$

$x^{4}$

$4 x^{3}$

$0 x^{2}$

$20 x$

$80$

$x^{2}$

$4 x$

$5$

$x^{6}$

$0 x^{5}$

$21 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{6}$

$4 x^{5}$

$5 x^{4}$

$4 x^{5}$

$16 x^{4}$

$0 x^{3}$

$4 x^{5}$

$16 x^{4}$

$20 x^{3}$

$0 x^{2}$

$0 x$

$20 x^{3}$

$80 x^{2}$

$100 x$

$80 x^{2}$

$100 x$

$0$

$80 x^{2}$

$320 x$

$400$

Step 2.3.11.20

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{4}$

$4 x^{3}$

$0 x^{2}$

$20 x$

$80$

$x^{2}$

$4 x$

$5$

$x^{6}$

$0 x^{5}$

$21 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{6}$

$4 x^{5}$

$5 x^{4}$

$4 x^{5}$

$16 x^{4}$

$0 x^{3}$

$4 x^{5}$

$16 x^{4}$

$20 x^{3}$

$0 x^{2}$

$0 x$

$20 x^{3}$

$80 x^{2}$

$100 x$

$80 x^{2}$

$100 x$

$0$

$80 x^{2}$

$320 x$

$400$

$420 x$

$400$

Step 2.3.11.21

The final answer is the quotient plus the remainder over the divisor.

$y + C_{1} = \int - x^{4} - 4 x^{3} - 20 x - 80 + \frac{420 x + 400}{- x^{2} + 4 x + 5} d x$

Step 2.3.12

Split the single integral into multiple integrals.

$y + C_{1} = \int - x^{4} d x + \int - 4 x^{3} d x + \int - 20 x d x + \int - 80 d x + \int \frac{420 x + 400}{- x^{2} + 4 x + 5} d x$

Step 2.3.13

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$y + C_{1} = - \int x^{4} d x + \int - 4 x^{3} d x + \int - 20 x d x + \int - 80 d x + \int \frac{420 x + 400}{- x^{2} + 4 x + 5} d x$

Step 2.3.14

By the Power Rule, the integral of $x^{4}$ with respect to $x$ is $\frac{1}{5} x^{5}$ .

$y + C_{1} = - (\frac{1}{5} x^{5} + C_{2}) + \int - 4 x^{3} d x + \int - 20 x d x + \int - 80 d x + \int \frac{420 x + 400}{- x^{2} + 4 x + 5} d x$

Step 2.3.15

Since $- 4$ is constant with respect to $x$ , move $- 4$ out of the integral.

$y + C_{1} = - (\frac{1}{5} x^{5} + C_{2}) - 4 \int x^{3} d x + \int - 20 x d x + \int - 80 d x + \int \frac{420 x + 400}{- x^{2} + 4 x + 5} d x$

Step 2.3.16

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$y + C_{1} = - (\frac{1}{5} x^{5} + C_{2}) - 4 (\frac{1}{4} x^{4} + C_{3}) + \int - 20 x d x + \int - 80 d x + \int \frac{420 x + 400}{- x^{2} + 4 x + 5} d x$

Step 2.3.17

Since $- 20$ is constant with respect to $x$ , move $- 20$ out of the integral.

$y + C_{1} = - (\frac{1}{5} x^{5} + C_{2}) - 4 (\frac{1}{4} x^{4} + C_{3}) - 20 \int x d x + \int - 80 d x + \int \frac{420 x + 400}{- x^{2} + 4 x + 5} d x$

Step 2.3.18

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$y + C_{1} = - (\frac{1}{5} x^{5} + C_{2}) - 4 (\frac{1}{4} x^{4} + C_{3}) - 20 (\frac{1}{2} x^{2} + C_{4}) + \int - 80 d x + \int \frac{420 x + 400}{- x^{2} + 4 x + 5} d x$

Step 2.3.19

Apply the constant rule.

$y + C_{1} = - (\frac{1}{5} x^{5} + C_{2}) - 4 (\frac{1}{4} x^{4} + C_{3}) - 20 (\frac{1}{2} x^{2} + C_{4}) - 80 x + C_{5} + \int \frac{420 x + 400}{- x^{2} + 4 x + 5} d x$

Step 2.3.20

Simplify.

Tap for more steps...

Step 2.3.20.1

Combine $\frac{1}{5}$ and $x^{5}$ .

$y + C_{1} = - (\frac{x^{5}}{5} + C_{2}) - 4 (\frac{1}{4} x^{4} + C_{3}) - 20 (\frac{1}{2} x^{2} + C_{4}) - 80 x + C_{5} + \int \frac{420 x + 400}{- x^{2} + 4 x + 5} d x$

Step 2.3.20.2

Combine $\frac{1}{4}$ and $x^{4}$ .

$y + C_{1} = - (\frac{x^{5}}{5} + C_{2}) - 4 (\frac{x^{4}}{4} + C_{3}) - 20 (\frac{1}{2} x^{2} + C_{4}) - 80 x + C_{5} + \int \frac{420 x + 400}{- x^{2} + 4 x + 5} d x$

Step 2.3.21

Write the fraction using partial fraction decomposition.

Tap for more steps...

Step 2.3.21.1

Decompose the fraction and multiply through by the common denominator.

Tap for more steps...

Step 2.3.21.1.1

Factor the fraction.

Tap for more steps...

Step 2.3.21.1.1.1

Factor $20$ out of $420 x + 400$ .

Tap for more steps...

Step 2.3.21.1.1.1.1

Factor $20$ out of $420 x$ .

$\frac{20 (21 x) + 400}{- x^{2} + 4 x + 5}$

Step 2.3.21.1.1.1.2

Factor $20$ out of $400$ .

$\frac{20 (21 x) + 20 (20)}{- x^{2} + 4 x + 5}$

Step 2.3.21.1.1.1.3

Factor $20$ out of $20 (21 x) + 20 (20)$ .

$\frac{20 (21 x + 20)}{- x^{2} + 4 x + 5}$

Step 2.3.21.1.1.2

Factor by grouping.

Tap for more steps...

Step 2.3.21.1.1.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 5 = - 5$ and whose sum is $b = 4$ .

Tap for more steps...

Step 2.3.21.1.1.2.1.1

Factor $4$ out of $4 x$ .

$\frac{20 (21 x + 20)}{- x^{2} + 4 (x) + 5}$

Step 2.3.21.1.1.2.1.2

Rewrite $4$ as $- 1$ plus $5$

$\frac{20 (21 x + 20)}{- x^{2} + (- 1 + 5) x + 5}$

Step 2.3.21.1.1.2.1.3

Apply the distributive property.

$\frac{20 (21 x + 20)}{- x^{2} - 1 x + 5 x + 5}$

Step 2.3.21.1.1.2.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 2.3.21.1.1.2.2.1

Group the first two terms and the last two terms.

$\frac{20 (21 x + 20)}{(- x^{2} - 1 x) + 5 x + 5}$

Step 2.3.21.1.1.2.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{20 (21 x + 20)}{x (- x - 1) - 5 (- x - 1)}$

Step 2.3.21.1.1.2.3

Factor the polynomial by factoring out the greatest common factor, $- x - 1$ .

$\frac{20 (21 x + 20)}{(- x - 1) (x - 5)}$

Step 2.3.21.1.2

For each factor in the denominator, create a new fraction using the factor as the denominator, and an unknown value as the numerator. Since the factor in the denominator is linear, put a single variable in its place $A$ .

$\frac{A}{- x - 1}$

Step 2.3.21.1.3

$\frac{A}{- x - 1} + \frac{B}{x - 5}$

Step 2.3.21.1.4

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is $(- x - 1) (x - 5)$ .

$\frac{20 (21 x + 20) (- x - 1) (x - 5)}{(- x - 1) (x - 5)} = \frac{(A) (- x - 1) (x - 5)}{- x - 1} + \frac{(B) (- x - 1) (x - 5)}{x - 5}$

Step 2.3.21.1.5

Cancel the common factor of $- x - 1$ .

Tap for more steps...

Step 2.3.21.1.5.1

Cancel the common factor.

$\frac{20 (21 x + 20) (- x - 1) (x - 5)}{(- x - 1) (x - 5)} = \frac{(A) (- x - 1) (x - 5)}{- x - 1} + \frac{(B) (- x - 1) (x - 5)}{x - 5}$

Step 2.3.21.1.5.2

Rewrite the expression.

$\frac{20 (21 x + 20) (x - 5)}{x - 5} = \frac{(A) (- x - 1) (x - 5)}{- x - 1} + \frac{(B) (- x - 1) (x - 5)}{x - 5}$

Step 2.3.21.1.6

Cancel the common factor of $x - 5$ .

Tap for more steps...

Step 2.3.21.1.6.1

Cancel the common factor.

$\frac{20 (21 x + 20) (x - 5)}{x - 5} = \frac{(A) (- x - 1) (x - 5)}{- x - 1} + \frac{(B) (- x - 1) (x - 5)}{x - 5}$

Step 2.3.21.1.6.2

Divide $20 (21 x + 20)$ by $1$ .

$20 (21 x + 20) = \frac{(A) (- x - 1) (x - 5)}{- x - 1} + \frac{(B) (- x - 1) (x - 5)}{x - 5}$

Step 2.3.21.1.7

Apply the distributive property.

$20 (21 x) + 20 \cdot 20 = \frac{(A) (- x - 1) (x - 5)}{- x - 1} + \frac{(B) (- x - 1) (x - 5)}{x - 5}$

Step 2.3.21.1.8

Multiply.

Tap for more steps...

Step 2.3.21.1.8.1

Multiply $21$ by $20$ .

$420 x + 20 \cdot 20 = \frac{(A) (- x - 1) (x - 5)}{- x - 1} + \frac{(B) (- x - 1) (x - 5)}{x - 5}$

Step 2.3.21.1.8.2

Multiply $20$ by $20$ .

$420 x + 400 = \frac{(A) (- x - 1) (x - 5)}{- x - 1} + \frac{(B) (- x - 1) (x - 5)}{x - 5}$

Step 2.3.21.1.9

Simplify each term.

Tap for more steps...

Step 2.3.21.1.9.1

Cancel the common factor of $- x - 1$ .

Tap for more steps...

Step 2.3.21.1.9.1.1

Cancel the common factor.

$420 x + 400 = \frac{A (- x - 1) (x - 5)}{- x - 1} + \frac{(B) (- x - 1) (x - 5)}{x - 5}$

Step 2.3.21.1.9.1.2

Divide $(A) (x - 5)$ by $1$ .

$420 x + 400 = (A) (x - 5) + \frac{(B) (- x - 1) (x - 5)}{x - 5}$

Step 2.3.21.1.9.2

Apply the distributive property.

$420 x + 400 = A x + A \cdot - 5 + \frac{(B) (- x - 1) (x - 5)}{x - 5}$

Step 2.3.21.1.9.3

Move $- 5$ to the left of $A$ .

$420 x + 400 = A x - 5 \cdot A + \frac{(B) (- x - 1) (x - 5)}{x - 5}$

Step 2.3.21.1.9.4

Cancel the common factor of $x - 5$ .

Tap for more steps...

Step 2.3.21.1.9.4.1

Cancel the common factor.

$420 x + 400 = A x - 5 A + \frac{(B) (- x - 1) (x - 5)}{x - 5}$

Step 2.3.21.1.9.4.2

Divide $(B) (- x - 1)$ by $1$ .

$420 x + 400 = A x - 5 A + (B) (- x - 1)$

Step 2.3.21.1.9.5

Apply the distributive property.

$420 x + 400 = A x - 5 A + B (- x) + B \cdot - 1$

Step 2.3.21.1.9.6

Rewrite using the commutative property of multiplication.

$420 x + 400 = A x - 5 A - B x + B \cdot - 1$

Step 2.3.21.1.9.7

Move $- 1$ to the left of $B$ .

$420 x + 400 = A x - 5 A - B x - 1 \cdot B$

Step 2.3.21.1.9.8

Rewrite $- 1 B$ as $- B$ .

$420 x + 400 = A x - 5 A - B x - B$

Step 2.3.21.1.10

Simplify the expression.

Tap for more steps...

Step 2.3.21.1.10.1

Move $B$ .

$420 x + 400 = A x - 5 A - x B - B$

Step 2.3.21.1.10.2

Move $- 5 A$ .

$420 x + 400 = A x - x B - 5 A - B$

Step 2.3.21.2

Create equations for the partial fraction variables and use them to set up a system of equations.

Tap for more steps...

Step 2.3.21.2.1

Create an equation for the partial fraction variables by equating the coefficients of $x$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$420 = A - 1 B$

Step 2.3.21.2.2

Create an equation for the partial fraction variables by equating the coefficients of the terms not containing $x$ . For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$400 = - 5 A - 1 B$

Step 2.3.21.2.3

Set up the system of equations to find the coefficients of the partial fractions.

$420 = A - 1 B$

$400 = - 5 A - 1 B$

$420 = A - 1 B$

$400 = - 5 A - 1 B$

Step 2.3.21.3

Solve the system of equations.

Tap for more steps...

Step 2.3.21.3.1

Solve for $A$ in $420 = A - 1 B$ .

Tap for more steps...

Step 2.3.21.3.1.1

Rewrite the equation as $A - 1 B = 420$ .

$A - 1 B = 420$

$400 = - 5 A - 1 B$

Step 2.3.21.3.1.2

Rewrite $- 1 B$ as $- B$ .

$A - B = 420$

$400 = - 5 A - 1 B$

Step 2.3.21.3.1.3

Add $B$ to both sides of the equation.

$A = 420 + B$

$400 = - 5 A - 1 B$

$A = 420 + B$

$400 = - 5 A - 1 B$

Step 2.3.21.3.2

Replace all occurrences of $A$ with $420 + B$ in each equation.

Tap for more steps...

Step 2.3.21.3.2.1

Replace all occurrences of $A$ in $400 = - 5 A - 1 B$ with $420 + B$ .

$400 = - 5 (420 + B) - 1 B$

$A = 420 + B$

Step 2.3.21.3.2.2

Simplify the right side.

Tap for more steps...

Step 2.3.21.3.2.2.1

Simplify $- 5 (420 + B) - 1 B$ .

Tap for more steps...

Step 2.3.21.3.2.2.1.1

Simplify each term.

Tap for more steps...

Step 2.3.21.3.2.2.1.1.1

Apply the distributive property.

$400 = - 5 \cdot 420 - 5 B - 1 B$

$A = 420 + B$

Step 2.3.21.3.2.2.1.1.2

Multiply $- 5$ by $420$ .

$400 = - 2100 - 5 B - 1 B$

$A = 420 + B$

Step 2.3.21.3.2.2.1.1.3

Rewrite $- 1 B$ as $- B$ .

$400 = - 2100 - 5 B - B$

$A = 420 + B$

$400 = - 2100 - 5 B - B$

$A = 420 + B$

Step 2.3.21.3.2.2.1.2

Subtract $B$ from $- 5 B$ .

$400 = - 2100 - 6 B$

$A = 420 + B$

$400 = - 2100 - 6 B$

$A = 420 + B$

$400 = - 2100 - 6 B$

$A = 420 + B$

$400 = - 2100 - 6 B$

$A = 420 + B$

Step 2.3.21.3.3

Solve for $B$ in $400 = - 2100 - 6 B$ .

Tap for more steps...

Step 2.3.21.3.3.1

Rewrite the equation as $- 2100 - 6 B = 400$ .

$- 2100 - 6 B = 400$

$A = 420 + B$

Step 2.3.21.3.3.2

Move all terms not containing $B$ to the right side of the equation.

Tap for more steps...

Step 2.3.21.3.3.2.1

Add $2100$ to both sides of the equation.

$- 6 B = 400 + 2100$

$A = 420 + B$

Step 2.3.21.3.3.2.2

Add $400$ and $2100$ .

$- 6 B = 2500$

$A = 420 + B$

$- 6 B = 2500$

$A = 420 + B$

Step 2.3.21.3.3.3

Divide each term in $- 6 B = 2500$ by $- 6$ and simplify.

Tap for more steps...

Step 2.3.21.3.3.3.1

Divide each term in $- 6 B = 2500$ by $- 6$ .

$\frac{- 6 B}{- 6} = \frac{2500}{- 6}$

$A = 420 + B$

Step 2.3.21.3.3.3.2

Simplify the left side.

Tap for more steps...

Step 2.3.21.3.3.3.2.1

Cancel the common factor of $- 6$ .

Tap for more steps...

Step 2.3.21.3.3.3.2.1.1

Cancel the common factor.

$\frac{- 6 B}{- 6} = \frac{2500}{- 6}$

$A = 420 + B$

Step 2.3.21.3.3.3.2.1.2

Divide $B$ by $1$ .

$B = \frac{2500}{- 6}$

$A = 420 + B$

$B = \frac{2500}{- 6}$

$A = 420 + B$

$B = \frac{2500}{- 6}$

$A = 420 + B$

Step 2.3.21.3.3.3.3

Simplify the right side.

Tap for more steps...

Step 2.3.21.3.3.3.3.1

Cancel the common factor of $2500$ and $- 6$ .

Tap for more steps...

Step 2.3.21.3.3.3.3.1.1

Factor $2$ out of $2500$ .

$B = \frac{2 (1250)}{- 6}$

$A = 420 + B$

Step 2.3.21.3.3.3.3.1.2

Cancel the common factors.

Tap for more steps...

Step 2.3.21.3.3.3.3.1.2.1

Factor $2$ out of $- 6$ .

$B = \frac{2 \cdot 1250}{2 \cdot - 3}$

$A = 420 + B$

Step 2.3.21.3.3.3.3.1.2.2

Cancel the common factor.

$B = \frac{2 \cdot 1250}{2 \cdot - 3}$

$A = 420 + B$

Step 2.3.21.3.3.3.3.1.2.3

Rewrite the expression.

$B = \frac{1250}{- 3}$

$A = 420 + B$

$B = \frac{1250}{- 3}$

$A = 420 + B$

$B = \frac{1250}{- 3}$

$A = 420 + B$

Step 2.3.21.3.3.3.3.2

Move the negative in front of the fraction.

$B = - \frac{1250}{3}$

$A = 420 + B$

$B = - \frac{1250}{3}$

$A = 420 + B$

$B = - \frac{1250}{3}$

$A = 420 + B$

$B = - \frac{1250}{3}$

$A = 420 + B$

Step 2.3.21.3.4

Replace all occurrences of $B$ with $- \frac{1250}{3}$ in each equation.

Tap for more steps...

Step 2.3.21.3.4.1

Replace all occurrences of $B$ in $A = 420 + B$ with $- \frac{1250}{3}$ .

$A = 420 - \frac{1250}{3}$

$B = - \frac{1250}{3}$

Step 2.3.21.3.4.2

Simplify $A = 420 - \frac{1250}{3}$ .

Tap for more steps...

Step 2.3.21.3.4.2.1

Simplify the left side.

Tap for more steps...

Step 2.3.21.3.4.2.1.1

Remove parentheses.

$A = 420 - \frac{1250}{3}$

$B = - \frac{1250}{3}$

$A = 420 - \frac{1250}{3}$

$B = - \frac{1250}{3}$

Step 2.3.21.3.4.2.2

Simplify the right side.

Tap for more steps...

Step 2.3.21.3.4.2.2.1

Simplify $420 - \frac{1250}{3}$ .

Tap for more steps...

Step 2.3.21.3.4.2.2.1.1

To write $420$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$A = 420 \cdot \frac{3}{3} - \frac{1250}{3}$

$B = - \frac{1250}{3}$

Step 2.3.21.3.4.2.2.1.2

Combine $420$ and $\frac{3}{3}$ .

$A = \frac{420 \cdot 3}{3} - \frac{1250}{3}$

$B = - \frac{1250}{3}$

Step 2.3.21.3.4.2.2.1.3

Combine the numerators over the common denominator.

$A = \frac{420 \cdot 3 - 1250}{3}$

$B = - \frac{1250}{3}$

Step 2.3.21.3.4.2.2.1.4

Simplify the numerator.

Tap for more steps...

Step 2.3.21.3.4.2.2.1.4.1

Multiply $420$ by $3$ .

$A = \frac{1260 - 1250}{3}$

$B = - \frac{1250}{3}$

Step 2.3.21.3.4.2.2.1.4.2

Subtract $1250$ from $1260$ .

$A = \frac{10}{3}$

$B = - \frac{1250}{3}$

$A = \frac{10}{3}$

$B = - \frac{1250}{3}$

$A = \frac{10}{3}$

$B = - \frac{1250}{3}$

$A = \frac{10}{3}$

$B = - \frac{1250}{3}$

$A = \frac{10}{3}$

$B = - \frac{1250}{3}$

$A = \frac{10}{3}$

$B = - \frac{1250}{3}$

Step 2.3.21.3.5

List all of the solutions.

$A = \frac{10}{3}, B = - \frac{1250}{3}$

Step 2.3.21.4

Replace each of the partial fraction coefficients in $\frac{A}{- x - 1} + \frac{B}{x - 5}$ with the values found for $A$ and $B$ .

$\frac{\frac{10}{3}}{- x - 1} + \frac{- \frac{1250}{3}}{x - 5}$

Step 2.3.21.5

Simplify.

Tap for more steps...

Step 2.3.21.5.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{10}{3} \cdot \frac{1}{- x - 1} + \frac{- \frac{1250}{3}}{x - 5}$

Step 2.3.21.5.2

Multiply $\frac{10}{3}$ by $\frac{1}{- x - 1}$ .

$\frac{10}{3 (- x - 1)} + \frac{- \frac{1250}{3}}{x - 5}$

Step 2.3.21.5.3

Factor $- 1$ out of $- x$ .

$\frac{10}{3 (- (x) - 1)} + \frac{- \frac{1250}{3}}{x - 5}$

Step 2.3.21.5.4

Rewrite $- 1$ as $- 1 (1)$ .

$\frac{10}{3 (- (x) - 1 \cdot 1)} + \frac{- \frac{1250}{3}}{x - 5}$

Step 2.3.21.5.5

Factor $- 1$ out of $- (x) - 1 (1)$ .

$\frac{10}{3 (- (x + 1))} + \frac{- \frac{1250}{3}}{x - 5}$

Step 2.3.21.5.6

Rewrite negatives.

Tap for more steps...

Step 2.3.21.5.6.1

Rewrite $- (x + 1)$ as $- 1 (x + 1)$ .

$\frac{10}{3 (- 1 (x + 1))} + \frac{- \frac{1250}{3}}{x - 5}$

Step 2.3.21.5.6.2

Move the negative in front of the fraction.

$- \frac{10}{3 (x + 1)} + \frac{- \frac{1250}{3}}{x - 5}$

Step 2.3.21.5.7

Multiply the numerator by the reciprocal of the denominator.

$- \frac{10}{3 (x + 1)} - \frac{1250}{3} \cdot \frac{1}{x - 5}$

Step 2.3.21.5.8

Multiply $\frac{1}{x - 5}$ by $\frac{1250}{3}$ .

$- \frac{10}{3 (x + 1)} - \frac{1250}{(x - 5) \cdot 3}$

Step 2.3.21.5.9

Move $3$ to the left of $x - 5$ .

$y + C_{1} = - (\frac{x^{5}}{5} + C_{2}) - 4 (\frac{x^{4}}{4} + C_{3}) - 20 (\frac{1}{2} x^{2} + C_{4}) - 80 x + C_{5} + \int - \frac{10}{3 (x + 1)} - \frac{1250}{3 (x - 5)} d x$

Step 2.3.22

Split the single integral into multiple integrals.

$y + C_{1} = - (\frac{x^{5}}{5} + C_{2}) - 4 (\frac{x^{4}}{4} + C_{3}) - 20 (\frac{1}{2} x^{2} + C_{4}) - 80 x + C_{5} + \int - \frac{10}{3 (x + 1)} d x + \int - \frac{1250}{3 (x - 5)} d x$

Step 2.3.23

Combine $\frac{1}{2}$ and $x^{2}$ .

$y + C_{1} = - (\frac{x^{5}}{5} + C_{2}) - 4 (\frac{x^{4}}{4} + C_{3}) - 20 (\frac{x^{2}}{2} + C_{4}) - 80 x + C_{5} + \int - \frac{10}{3 (x + 1)} d x + \int - \frac{1250}{3 (x - 5)} d x$

Step 2.3.24

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$y + C_{1} = - (\frac{x^{5}}{5} + C_{2}) - 4 (\frac{x^{4}}{4} + C_{3}) - 20 (\frac{x^{2}}{2} + C_{4}) - 80 x + C_{5} - \int \frac{10}{3 (x + 1)} d x + \int - \frac{1250}{3 (x - 5)} d x$

Step 2.3.25

Since $\frac{10}{3}$ is constant with respect to $x$ , move $\frac{10}{3}$ out of the integral.

$y + C_{1} = - (\frac{x^{5}}{5} + C_{2}) - 4 (\frac{x^{4}}{4} + C_{3}) - 20 (\frac{x^{2}}{2} + C_{4}) - 80 x + C_{5} - (\frac{10}{3} \int \frac{1}{x + 1} d x) + \int - \frac{1250}{3 (x - 5)} d x$

Step 2.3.26

Let $u_{1} = x + 1$ . Then $d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

Tap for more steps...

Step 2.3.26.1

Let $u_{1} = x + 1$ . Find $\frac{d u_{1}}{d x}$ .

Tap for more steps...

Step 2.3.26.1.1

Differentiate $x + 1$ .

$\frac{d}{d x} [x + 1]$

Step 2.3.26.1.2

By the Sum Rule, the derivative of $x + 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 2.3.26.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d x} [1]$

Step 2.3.26.1.4

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$1 + 0$

Step 2.3.26.1.5

Add $1$ and $0$ .

$1$

Step 2.3.26.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$y + C_{1} = - (\frac{x^{5}}{5} + C_{2}) - 4 (\frac{x^{4}}{4} + C_{3}) - 20 (\frac{x^{2}}{2} + C_{4}) - 80 x + C_{5} - \frac{10}{3} \int \frac{1}{u_{1}} d u_{1} + \int - \frac{1250}{3 (x - 5)} d x$

Step 2.3.27

The integral of $\frac{1}{u_{1}}$ with respect to $u_{1}$ is $\ln (| u_{1} |)$ .

$y + C_{1} = - (\frac{x^{5}}{5} + C_{2}) - 4 (\frac{x^{4}}{4} + C_{3}) - 20 (\frac{x^{2}}{2} + C_{4}) - 80 x + C_{5} - \frac{10}{3} (\ln (| u_{1} |) + C_{6}) + \int - \frac{1250}{3 (x - 5)} d x$

Step 2.3.28

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$y + C_{1} = - (\frac{x^{5}}{5} + C_{2}) - 4 (\frac{x^{4}}{4} + C_{3}) - 20 (\frac{x^{2}}{2} + C_{4}) - 80 x + C_{5} - \frac{10}{3} (\ln (| u_{1} |) + C_{6}) - \int \frac{1250}{3 (x - 5)} d x$

Step 2.3.29

Since $\frac{1250}{3}$ is constant with respect to $x$ , move $\frac{1250}{3}$ out of the integral.

$y + C_{1} = - (\frac{x^{5}}{5} + C_{2}) - 4 (\frac{x^{4}}{4} + C_{3}) - 20 (\frac{x^{2}}{2} + C_{4}) - 80 x + C_{5} - \frac{10}{3} (\ln (| u_{1} |) + C_{6}) - (\frac{1250}{3} \int \frac{1}{x - 5} d x)$

Step 2.3.30

Let $u_{2} = x - 5$ . Then $d u_{2} = d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

Tap for more steps...

Step 2.3.30.1

Let $u_{2} = x - 5$ . Find $\frac{d u_{2}}{d x}$ .

Tap for more steps...

Step 2.3.30.1.1

Differentiate $x - 5$ .

$\frac{d}{d x} [x - 5]$

Step 2.3.30.1.2

By the Sum Rule, the derivative of $x - 5$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 5]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- 5]$

Step 2.3.30.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d x} [- 5]$

Step 2.3.30.1.4

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$1 + 0$

Step 2.3.30.1.5

Add $1$ and $0$ .

$1$

Step 2.3.30.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$y + C_{1} = - (\frac{x^{5}}{5} + C_{2}) - 4 (\frac{x^{4}}{4} + C_{3}) - 20 (\frac{x^{2}}{2} + C_{4}) - 80 x + C_{5} - \frac{10}{3} (\ln (| u_{1} |) + C_{6}) - \frac{1250}{3} \int \frac{1}{u_{2}} d u_{2}$

Step 2.3.31

The integral of $\frac{1}{u_{2}}$ with respect to $u_{2}$ is $\ln (| u_{2} |)$ .

$y + C_{1} = - (\frac{x^{5}}{5} + C_{2}) - 4 (\frac{x^{4}}{4} + C_{3}) - 20 (\frac{x^{2}}{2} + C_{4}) - 80 x + C_{5} - \frac{10}{3} (\ln (| u_{1} |) + C_{6}) - \frac{1250}{3} (\ln (| u_{2} |) + C_{7})$

Step 2.3.32

Simplify.

$y + C_{1} = - \frac{x^{5}}{5} - x^{4} - 10 x^{2} - 80 x - \frac{10 \ln (| u_{1} |)}{3} - \frac{1250}{3} \ln (| u_{2} |) + C_{8}$

Step 2.3.33

Substitute back in for each integration substitution variable.

Tap for more steps...

Step 2.3.33.1

Replace all occurrences of $u_{1}$ with $x + 1$ .

$y + C_{1} = - \frac{x^{5}}{5} - x^{4} - 10 x^{2} - 80 x - \frac{10 \ln (| x + 1 |)}{3} - \frac{1250}{3} \ln (| u_{2} |) + C_{8}$

Step 2.3.33.2

Replace all occurrences of $u_{2}$ with $x - 5$ .

$y + C_{1} = - \frac{x^{5}}{5} - x^{4} - 10 x^{2} - 80 x - \frac{10 \ln (| x + 1 |)}{3} - \frac{1250}{3} \ln (| x - 5 |) + C_{8}$

Step 2.3.34

Reorder terms.

$y + C_{1} = - \frac{1}{5} x^{5} - x^{4} - 10 x^{2} - 80 x - \frac{10}{3} \ln (| x + 1 |) - \frac{1250}{3} \ln (| x - 5 |) + C_{8}$

Step 2.3.35

Reorder terms.

$y + C_{1} = - x^{4} - 10 x^{2} - \frac{1}{5} x^{5} - \frac{10}{3} \ln (| x + 1 |) - \frac{1250}{3} \ln (| x - 5 |) - 80 x + C_{8}$

Step 2.4

Group the constant of integration on the right side as $C$ .

$y = - x^{4} - 10 x^{2} - \frac{1}{5} x^{5} - \frac{10}{3} \ln (| x + 1 |) - \frac{1250}{3} \ln (| x - 5 |) - 80 x + C$