Calculus Examples

$\frac{2 x^{4} - 3 x^{2} + 2}{x^{2} - 6 x + 8}$

Step 1

Divide $2 x^{4} - 3 x^{2} + 2$ by $x^{2} - 6 x + 8$ .

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Step 1.1

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

$x^{2}$

$6 x$

$8$

$2 x^{4}$

$0 x^{3}$

$3 x^{2}$

$0 x$

$2$

Step 1.2

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

$2 x^{2}$

$x^{2}$

$6 x$

$8$

$2 x^{4}$

$0 x^{3}$

$3 x^{2}$

$0 x$

$2$

Step 1.3

Multiply the new quotient term by the divisor.

$2 x^{2}$

$x^{2}$

$6 x$

$8$

$2 x^{4}$

$0 x^{3}$

$3 x^{2}$

$0 x$

$2$

$2 x^{4}$

$12 x^{3}$

$16 x^{2}$

Step 1.4

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{4} - 12 x^{3} + 16 x^{2}$

$2 x^{2}$

$x^{2}$

$6 x$

$8$

$2 x^{4}$

$0 x^{3}$

$3 x^{2}$

$0 x$

$2$

$2 x^{4}$

$12 x^{3}$

$16 x^{2}$

Step 1.5

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

$2 x^{2}$

$x^{2}$

$6 x$

$8$

$2 x^{4}$

$0 x^{3}$

$3 x^{2}$

$0 x$

$2$

$2 x^{4}$

$12 x^{3}$

$16 x^{2}$

$12 x^{3}$

$19 x^{2}$

Step 1.6

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

$2 x^{2}$

$x^{2}$

$6 x$

$8$

$2 x^{4}$

$0 x^{3}$

$3 x^{2}$

$0 x$

$2$

$2 x^{4}$

$12 x^{3}$

$16 x^{2}$

$12 x^{3}$

$19 x^{2}$

$0 x$

Step 1.7

Divide the highest order term in the dividend $12 x^{3}$ by the highest order term in divisor $x^{2}$ .

$2 x^{2}$

$12 x$

$x^{2}$

$6 x$

$8$

$2 x^{4}$

$0 x^{3}$

$3 x^{2}$

$0 x$

$2$

$2 x^{4}$

$12 x^{3}$

$16 x^{2}$

$12 x^{3}$

$19 x^{2}$

$0 x$

Step 1.8

Multiply the new quotient term by the divisor.

$2 x^{2}$

$12 x$

$x^{2}$

$6 x$

$8$

$2 x^{4}$

$0 x^{3}$

$3 x^{2}$

$0 x$

$2$

$2 x^{4}$

$12 x^{3}$

$16 x^{2}$

$12 x^{3}$

$19 x^{2}$

$0 x$

$12 x^{3}$

$72 x^{2}$

$96 x$

Step 1.9

The expression needs to be subtracted from the dividend, so change all the signs in $12 x^{3} - 72 x^{2} + 96 x$

$2 x^{2}$

$12 x$

$x^{2}$

$6 x$

$8$

$2 x^{4}$

$0 x^{3}$

$3 x^{2}$

$0 x$

$2$

$2 x^{4}$

$12 x^{3}$

$16 x^{2}$

$12 x^{3}$

$19 x^{2}$

$0 x$

$12 x^{3}$

$72 x^{2}$

$96 x$

Step 1.10

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

$2 x^{2}$

$12 x$

$x^{2}$

$6 x$

$8$

$2 x^{4}$

$0 x^{3}$

$3 x^{2}$

$0 x$

$2$

$2 x^{4}$

$12 x^{3}$

$16 x^{2}$

$12 x^{3}$

$19 x^{2}$

$0 x$

$12 x^{3}$

$72 x^{2}$

$96 x$

$53 x^{2}$

$96 x$

Step 1.11

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

$2 x^{2}$

$12 x$

$x^{2}$

$6 x$

$8$

$2 x^{4}$

$0 x^{3}$

$3 x^{2}$

$0 x$

$2$

$2 x^{4}$

$12 x^{3}$

$16 x^{2}$

$12 x^{3}$

$19 x^{2}$

$0 x$

$12 x^{3}$

$72 x^{2}$

$96 x$

$53 x^{2}$

$96 x$

$2$

Step 1.12

Divide the highest order term in the dividend $53 x^{2}$ by the highest order term in divisor $x^{2}$ .

$2 x^{2}$

$12 x$

$53$

$x^{2}$

$6 x$

$8$

$2 x^{4}$

$0 x^{3}$

$3 x^{2}$

$0 x$

$2$

$2 x^{4}$

$12 x^{3}$

$16 x^{2}$

$12 x^{3}$

$19 x^{2}$

$0 x$

$12 x^{3}$

$72 x^{2}$

$96 x$

$53 x^{2}$

$96 x$

$2$

Step 1.13

Multiply the new quotient term by the divisor.

$2 x^{2}$

$12 x$

$53$

$x^{2}$

$6 x$

$8$

$2 x^{4}$

$0 x^{3}$

$3 x^{2}$

$0 x$

$2$

$2 x^{4}$

$12 x^{3}$

$16 x^{2}$

$12 x^{3}$

$19 x^{2}$

$0 x$

$12 x^{3}$

$72 x^{2}$

$96 x$

$53 x^{2}$

$96 x$

$2$

$53 x^{2}$

$318 x$

$424$

Step 1.14

The expression needs to be subtracted from the dividend, so change all the signs in $53 x^{2} - 318 x + 424$

$2 x^{2}$

$12 x$

$53$

$x^{2}$

$6 x$

$8$

$2 x^{4}$

$0 x^{3}$

$3 x^{2}$

$0 x$

$2$

$2 x^{4}$

$12 x^{3}$

$16 x^{2}$

$12 x^{3}$

$19 x^{2}$

$0 x$

$12 x^{3}$

$72 x^{2}$

$96 x$

$53 x^{2}$

$96 x$

$2$

$53 x^{2}$

$318 x$

$424$

Step 1.15

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

$2 x^{2}$

$12 x$

$53$

$x^{2}$

$6 x$

$8$

$2 x^{4}$

$0 x^{3}$

$3 x^{2}$

$0 x$

$2$

$2 x^{4}$

$12 x^{3}$

$16 x^{2}$

$12 x^{3}$

$19 x^{2}$

$0 x$

$12 x^{3}$

$72 x^{2}$

$96 x$

$53 x^{2}$

$96 x$

$2$

$53 x^{2}$

$318 x$

$424$

$222 x$

$422$

Step 1.16

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

$\int 2 x^{2} + 12 x + 53 + \frac{222 x - 422}{x^{2} - 6 x + 8} d x$

Step 2

Split the single integral into multiple integrals.

$\int 2 x^{2} d x + \int 12 x d x + \int 53 d x + \int \frac{222 x - 422}{x^{2} - 6 x + 8} d x$

Step 3

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int x^{2} d x + \int 12 x d x + \int 53 d x + \int \frac{222 x - 422}{x^{2} - 6 x + 8} d x$

Step 4

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$2 (\frac{1}{3} x^{3} + C) + \int 12 x d x + \int 53 d x + \int \frac{222 x - 422}{x^{2} - 6 x + 8} d x$

Step 5

Since $12$ is constant with respect to $x$ , move $12$ out of the integral.

$2 (\frac{1}{3} x^{3} + C) + 12 \int x d x + \int 53 d x + \int \frac{222 x - 422}{x^{2} - 6 x + 8} d x$

Step 6

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

$2 (\frac{1}{3} x^{3} + C) + 12 (\frac{1}{2} x^{2} + C) + \int 53 d x + \int \frac{222 x - 422}{x^{2} - 6 x + 8} d x$

Step 7

Apply the constant rule.

$2 (\frac{1}{3} x^{3} + C) + 12 (\frac{1}{2} x^{2} + C) + 53 x + C + \int \frac{222 x - 422}{x^{2} - 6 x + 8} d x$

Step 8

Simplify.

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Step 8.1

Combine $\frac{1}{3}$ and $x^{3}$ .

$2 (\frac{x^{3}}{3} + C) + 12 (\frac{1}{2} x^{2} + C) + 53 x + C + \int \frac{222 x - 422}{x^{2} - 6 x + 8} d x$

Step 8.2

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

$2 (\frac{x^{3}}{3} + C) + 12 (\frac{x^{2}}{2} + C) + 53 x + C + \int \frac{222 x - 422}{x^{2} - 6 x + 8} d x$

Step 9

Write the fraction using partial fraction decomposition.

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Step 9.1

Decompose the fraction and multiply through by the common denominator.

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Step 9.1.1

Factor the fraction.

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Step 9.1.1.1

Factor $2$ out of $222 x - 422$ .

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Step 9.1.1.1.1

Factor $2$ out of $222 x$ .

$\frac{2 (111 x) - 422}{x^{2} - 6 x + 8}$

Step 9.1.1.1.2

Factor $2$ out of $- 422$ .

$\frac{2 (111 x) + 2 (- 211)}{x^{2} - 6 x + 8}$

Step 9.1.1.1.3

Factor $2$ out of $2 (111 x) + 2 (- 211)$ .

$\frac{2 (111 x - 211)}{x^{2} - 6 x + 8}$

Step 9.1.1.2

Factor $x^{2} - 6 x + 8$ using the AC method.

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Step 9.1.1.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $8$ and whose sum is $- 6$ .

$- 4, - 2$

Step 9.1.1.2.2

Write the factored form using these integers.

$\frac{2 (111 x - 211)}{(x - 4) (x - 2)}$

Step 9.1.2

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 $A$ .

$\frac{A}{x - 4}$

Step 9.1.3

$\frac{A}{x - 4} + \frac{B}{x - 2}$

Step 9.1.4

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is $(x - 4) (x - 2)$ .

$\frac{2 (111 x - 211) (x - 4) (x - 2)}{(x - 4) (x - 2)} = \frac{(A) (x - 4) (x - 2)}{x - 4} + \frac{(B) (x - 4) (x - 2)}{x - 2}$

Step 9.1.5

Cancel the common factor of $x - 4$ .

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Step 9.1.5.1

Cancel the common factor.

$\frac{2 (111 x - 211) (x - 4) (x - 2)}{(x - 4) (x - 2)} = \frac{(A) (x - 4) (x - 2)}{x - 4} + \frac{(B) (x - 4) (x - 2)}{x - 2}$

Step 9.1.5.2

Rewrite the expression.

$\frac{2 (111 x - 211) (x - 2)}{x - 2} = \frac{(A) (x - 4) (x - 2)}{x - 4} + \frac{(B) (x - 4) (x - 2)}{x - 2}$

Step 9.1.6

Cancel the common factor of $x - 2$ .

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Step 9.1.6.1

Cancel the common factor.

$\frac{2 (111 x - 211) (x - 2)}{x - 2} = \frac{(A) (x - 4) (x - 2)}{x - 4} + \frac{(B) (x - 4) (x - 2)}{x - 2}$

Step 9.1.6.2

Divide $2 (111 x - 211)$ by $1$ .

$2 (111 x - 211) = \frac{(A) (x - 4) (x - 2)}{x - 4} + \frac{(B) (x - 4) (x - 2)}{x - 2}$

Step 9.1.7

Apply the distributive property.

$2 (111 x) + 2 \cdot - 211 = \frac{(A) (x - 4) (x - 2)}{x - 4} + \frac{(B) (x - 4) (x - 2)}{x - 2}$

Step 9.1.8

Multiply.

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Step 9.1.8.1

Multiply $111$ by $2$ .

$222 x + 2 \cdot - 211 = \frac{(A) (x - 4) (x - 2)}{x - 4} + \frac{(B) (x - 4) (x - 2)}{x - 2}$

Step 9.1.8.2

Multiply $2$ by $- 211$ .

$222 x - 422 = \frac{(A) (x - 4) (x - 2)}{x - 4} + \frac{(B) (x - 4) (x - 2)}{x - 2}$

Step 9.1.9

Simplify each term.

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Step 9.1.9.1

Cancel the common factor of $x - 4$ .

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Step 9.1.9.1.1

Cancel the common factor.

$222 x - 422 = \frac{A (x - 4) (x - 2)}{x - 4} + \frac{(B) (x - 4) (x - 2)}{x - 2}$

Step 9.1.9.1.2

Divide $(A) (x - 2)$ by $1$ .

$222 x - 422 = (A) (x - 2) + \frac{(B) (x - 4) (x - 2)}{x - 2}$

Step 9.1.9.2

Apply the distributive property.

$222 x - 422 = A x + A \cdot - 2 + \frac{(B) (x - 4) (x - 2)}{x - 2}$

Step 9.1.9.3

Move $- 2$ to the left of $A$ .

$222 x - 422 = A x - 2 \cdot A + \frac{(B) (x - 4) (x - 2)}{x - 2}$

Step 9.1.9.4

Cancel the common factor of $x - 2$ .

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Step 9.1.9.4.1

Cancel the common factor.

$222 x - 422 = A x - 2 A + \frac{(B) (x - 4) (x - 2)}{x - 2}$

Step 9.1.9.4.2

Divide $(B) (x - 4)$ by $1$ .

$222 x - 422 = A x - 2 A + (B) (x - 4)$

Step 9.1.9.5

Apply the distributive property.

$222 x - 422 = A x - 2 A + B x + B \cdot - 4$

Step 9.1.9.6

Move $- 4$ to the left of $B$ .

$222 x - 422 = A x - 2 A + B x - 4 B$

Step 9.1.10

Move $- 2 A$ .

$222 x - 422 = A x + B x - 2 A - 4 B$

Step 9.2

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

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Step 9.2.1

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.

$222 = A + B$

Step 9.2.2

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.

$- 422 = - 2 A - 4 B$

Step 9.2.3

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

$222 = A + B$

$- 422 = - 2 A - 4 B$

$222 = A + B$

$- 422 = - 2 A - 4 B$

Step 9.3

Solve the system of equations.

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Step 9.3.1

Solve for $A$ in $222 = A + B$ .

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Step 9.3.1.1

Rewrite the equation as $A + B = 222$ .

$A + B = 222$

$- 422 = - 2 A - 4 B$

Step 9.3.1.2

Subtract $B$ from both sides of the equation.

$A = 222 - B$

$- 422 = - 2 A - 4 B$

$A = 222 - B$

$- 422 = - 2 A - 4 B$

Step 9.3.2

Replace all occurrences of $A$ with $222 - B$ in each equation.

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Step 9.3.2.1

Replace all occurrences of $A$ in $- 422 = - 2 A - 4 B$ with $222 - B$ .

$- 422 = - 2 (222 - B) - 4 B$

$A = 222 - B$

Step 9.3.2.2

Simplify the right side.

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Step 9.3.2.2.1

Simplify $- 2 (222 - B) - 4 B$ .

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Step 9.3.2.2.1.1

Simplify each term.

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Step 9.3.2.2.1.1.1

Apply the distributive property.

$- 422 = - 2 \cdot 222 - 2 (- B) - 4 B$

$A = 222 - B$

Step 9.3.2.2.1.1.2

Multiply $- 2$ by $222$ .

$- 422 = - 444 - 2 (- B) - 4 B$

$A = 222 - B$

Step 9.3.2.2.1.1.3

Multiply $- 1$ by $- 2$ .

$- 422 = - 444 + 2 B - 4 B$

$A = 222 - B$

$- 422 = - 444 + 2 B - 4 B$

$A = 222 - B$

Step 9.3.2.2.1.2

Subtract $4 B$ from $2 B$ .

$- 422 = - 444 - 2 B$

$A = 222 - B$

$- 422 = - 444 - 2 B$

$A = 222 - B$

$- 422 = - 444 - 2 B$

$A = 222 - B$

$- 422 = - 444 - 2 B$

$A = 222 - B$

Step 9.3.3

Solve for $B$ in $- 422 = - 444 - 2 B$ .

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Step 9.3.3.1

Rewrite the equation as $- 444 - 2 B = - 422$ .

$- 444 - 2 B = - 422$

$A = 222 - B$

Step 9.3.3.2

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

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Step 9.3.3.2.1

Add $444$ to both sides of the equation.

$- 2 B = - 422 + 444$

$A = 222 - B$

Step 9.3.3.2.2

Add $- 422$ and $444$ .

$- 2 B = 22$

$A = 222 - B$

$- 2 B = 22$

$A = 222 - B$

Step 9.3.3.3

Divide each term in $- 2 B = 22$ by $- 2$ and simplify.

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Step 9.3.3.3.1

Divide each term in $- 2 B = 22$ by $- 2$ .

$\frac{- 2 B}{- 2} = \frac{22}{- 2}$

$A = 222 - B$

Step 9.3.3.3.2

Simplify the left side.

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Step 9.3.3.3.2.1

Cancel the common factor of $- 2$ .

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Step 9.3.3.3.2.1.1

Cancel the common factor.

$\frac{- 2 B}{- 2} = \frac{22}{- 2}$

$A = 222 - B$

Step 9.3.3.3.2.1.2

Divide $B$ by $1$ .

$B = \frac{22}{- 2}$

$A = 222 - B$

$B = \frac{22}{- 2}$

$A = 222 - B$

$B = \frac{22}{- 2}$

$A = 222 - B$

Step 9.3.3.3.3

Simplify the right side.

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Step 9.3.3.3.3.1

Divide $22$ by $- 2$ .

$B = - 11$

$A = 222 - B$

$B = - 11$

$A = 222 - B$

$B = - 11$

$A = 222 - B$

$B = - 11$

$A = 222 - B$

Step 9.3.4

Replace all occurrences of $B$ with $- 11$ in each equation.

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Step 9.3.4.1

Replace all occurrences of $B$ in $A = 222 - B$ with $- 11$ .

$A = 222 - (- 11)$

$B = - 11$

Step 9.3.4.2

Simplify the right side.

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Step 9.3.4.2.1

Simplify $222 - (- 11)$ .

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Step 9.3.4.2.1.1

Multiply $- 1$ by $- 11$ .

$A = 222 + 11$

$B = - 11$

Step 9.3.4.2.1.2

Add $222$ and $11$ .

$A = 233$

$B = - 11$

$A = 233$

$B = - 11$

$A = 233$

$B = - 11$

$A = 233$

$B = - 11$

Step 9.3.5

List all of the solutions.

$A = 233, B = - 11$

Step 9.4

Replace each of the partial fraction coefficients in $\frac{A}{x - 4} + \frac{B}{x - 2}$ with the values found for $A$ and $B$ .

$\frac{233}{x - 4} + \frac{- 11}{x - 2}$

Step 9.5

Move the negative in front of the fraction.

$2 (\frac{x^{3}}{3} + C) + 12 (\frac{x^{2}}{2} + C) + 53 x + C + \int \frac{233}{x - 4} - \frac{11}{x - 2} d x$

Step 10

Split the single integral into multiple integrals.

$2 (\frac{x^{3}}{3} + C) + 12 (\frac{x^{2}}{2} + C) + 53 x + C + \int \frac{233}{x - 4} d x + \int - \frac{11}{x - 2} d x$

Step 11

Since $233$ is constant with respect to $x$ , move $233$ out of the integral.

$2 (\frac{x^{3}}{3} + C) + 12 (\frac{x^{2}}{2} + C) + 53 x + C + 233 \int \frac{1}{x - 4} d x + \int - \frac{11}{x - 2} d x$

Step 12

Let $u_{1} = x - 4$ . Then $d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 12.1

Let $u_{1} = x - 4$ . Find $\frac{d u_{1}}{d x}$ .

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Step 12.1.1

Differentiate $x - 4$ .

$\frac{d}{d x} [x - 4]$

Step 12.1.2

By the Sum Rule, the derivative of $x - 4$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 4]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- 4]$

Step 12.1.3

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

$1 + \frac{d}{d x} [- 4]$

Step 12.1.4

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$1 + 0$

Step 12.1.5

Add $1$ and $0$ .

$1$

Step 12.2

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

$2 (\frac{x^{3}}{3} + C) + 12 (\frac{x^{2}}{2} + C) + 53 x + C + 233 \int \frac{1}{u_{1}} d u_{1} + \int - \frac{11}{x - 2} d x$

Step 13

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

$2 (\frac{x^{3}}{3} + C) + 12 (\frac{x^{2}}{2} + C) + 53 x + C + 233 (\ln (| u_{1} |) + C) + \int - \frac{11}{x - 2} d x$

Step 14

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

$2 (\frac{x^{3}}{3} + C) + 12 (\frac{x^{2}}{2} + C) + 53 x + C + 233 (\ln (| u_{1} |) + C) - \int \frac{11}{x - 2} d x$

Step 15

Since $11$ is constant with respect to $x$ , move $11$ out of the integral.

$2 (\frac{x^{3}}{3} + C) + 12 (\frac{x^{2}}{2} + C) + 53 x + C + 233 (\ln (| u_{1} |) + C) - (11 \int \frac{1}{x - 2} d x)$

Step 16

Multiply $11$ by $- 1$ .

$2 (\frac{x^{3}}{3} + C) + 12 (\frac{x^{2}}{2} + C) + 53 x + C + 233 (\ln (| u_{1} |) + C) - 11 \int \frac{1}{x - 2} d x$

Step 17

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

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Step 17.1

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

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Step 17.1.1

Differentiate $x - 2$ .

$\frac{d}{d x} [x - 2]$

Step 17.1.2

By the Sum Rule, the derivative of $x - 2$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 2]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- 2]$

Step 17.1.3

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

$1 + \frac{d}{d x} [- 2]$

Step 17.1.4

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$1 + 0$

Step 17.1.5

Add $1$ and $0$ .

$1$

Step 17.2

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

$2 (\frac{x^{3}}{3} + C) + 12 (\frac{x^{2}}{2} + C) + 53 x + C + 233 (\ln (| u_{1} |) + C) - 11 \int \frac{1}{u_{2}} d u_{2}$

Step 18

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

$2 (\frac{x^{3}}{3} + C) + 12 (\frac{x^{2}}{2} + C) + 53 x + C + 233 (\ln (| u_{1} |) + C) - 11 (\ln (| u_{2} |) + C)$

Step 19

Simplify.

$\frac{2 x^{3}}{3} + 6 x^{2} + 53 x + 233 \ln (| u_{1} |) - 11 \ln (| u_{2} |) + C$

Step 20

Substitute back in for each integration substitution variable.

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Step 20.1

Replace all occurrences of $u_{1}$ with $x - 4$ .

$\frac{2 x^{3}}{3} + 6 x^{2} + 53 x + 233 \ln (| x - 4 |) - 11 \ln (| u_{2} |) + C$

Step 20.2

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

$\frac{2 x^{3}}{3} + 6 x^{2} + 53 x + 233 \ln (| x - 4 |) - 11 \ln (| x - 2 |) + C$

Step 21

Reorder terms.

$\frac{2}{3} x^{3} + 6 x^{2} + 53 x + 233 \ln (| x - 4 |) - 11 \ln (| x - 2 |) + C$