Calculus Examples

$(x^{2} + 2 x - 3) \frac{d y}{d x} = x + 5$

Step 1

Separate the variables.

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Step 1.1

Divide each term in $(x^{2} + 2 x - 3) \frac{d y}{d x} = x + 5$ by $x^{2} + 2 x - 3$ and simplify.

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Step 1.1.1

Divide each term in $(x^{2} + 2 x - 3) \frac{d y}{d x} = x + 5$ by $x^{2} + 2 x - 3$ .

$\frac{(x^{2} + 2 x - 3) \frac{d y}{d x}}{x^{2} + 2 x - 3} = \frac{x}{x^{2} + 2 x - 3} + \frac{5}{x^{2} + 2 x - 3}$

Step 1.1.2

Simplify the left side.

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Step 1.1.2.1

Cancel the common factor of $x^{2} + 2 x - 3$ .

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Step 1.1.2.1.1

Cancel the common factor.

$\frac{(x^{2} + 2 x - 3) \frac{d y}{d x}}{x^{2} + 2 x - 3} = \frac{x}{x^{2} + 2 x - 3} + \frac{5}{x^{2} + 2 x - 3}$

Step 1.1.2.1.2

Divide $\frac{d y}{d x}$ by $1$ .

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

Step 1.1.3

Simplify the right side.

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Step 1.1.3.1

Combine the numerators over the common denominator.

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

Step 1.1.3.2

Factor $x^{2} + 2 x - 3$ using the AC method.

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Step 1.1.3.2.1

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 $- 3$ and whose sum is $2$ .

$- 1, 3$

Step 1.1.3.2.2

Write the factored form using these integers.

$\frac{d y}{d x} = \frac{x + 5}{(x - 1) (x + 3)}$

Step 1.2

Rewrite the equation.

$d y = \frac{x + 5}{(x - 1) (x + 3)} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int \frac{x + 5}{(x - 1) (x + 3)} d x$

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Write the fraction using partial fraction decomposition.

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Step 2.3.1.1

Decompose the fraction and multiply through by the common denominator.

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Step 2.3.1.1.1

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 2.3.1.1.2

$\frac{A}{x - 1} + \frac{B}{x + 3}$

Step 2.3.1.1.3

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is $(x - 1) (x + 3)$ .

$\frac{(x + 5) (x - 1) (x + 3)}{(x - 1) (x + 3)} = \frac{(A) (x - 1) (x + 3)}{x - 1} + \frac{(B) (x - 1) (x + 3)}{x + 3}$

Step 2.3.1.1.4

Cancel the common factor of $x - 1$ .

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Step 2.3.1.1.4.1

Cancel the common factor.

$\frac{(x + 5) (x - 1) (x + 3)}{(x - 1) (x + 3)} = \frac{(A) (x - 1) (x + 3)}{x - 1} + \frac{(B) (x - 1) (x + 3)}{x + 3}$

Step 2.3.1.1.4.2

Rewrite the expression.

$\frac{(x + 5) (x + 3)}{x + 3} = \frac{(A) (x - 1) (x + 3)}{x - 1} + \frac{(B) (x - 1) (x + 3)}{x + 3}$

Step 2.3.1.1.5

Cancel the common factor of $x + 3$ .

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Step 2.3.1.1.5.1

Cancel the common factor.

$\frac{(x + 5) (x + 3)}{x + 3} = \frac{(A) (x - 1) (x + 3)}{x - 1} + \frac{(B) (x - 1) (x + 3)}{x + 3}$

Step 2.3.1.1.5.2

Divide $x + 5$ by $1$ .

$x + 5 = \frac{(A) (x - 1) (x + 3)}{x - 1} + \frac{(B) (x - 1) (x + 3)}{x + 3}$

Step 2.3.1.1.6

Simplify each term.

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Step 2.3.1.1.6.1

Cancel the common factor of $x - 1$ .

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Step 2.3.1.1.6.1.1

Cancel the common factor.

$x + 5 = \frac{A (x - 1) (x + 3)}{x - 1} + \frac{(B) (x - 1) (x + 3)}{x + 3}$

Step 2.3.1.1.6.1.2

Divide $(A) (x + 3)$ by $1$ .

$x + 5 = (A) (x + 3) + \frac{(B) (x - 1) (x + 3)}{x + 3}$

Step 2.3.1.1.6.2

Apply the distributive property.

$x + 5 = A x + A \cdot 3 + \frac{(B) (x - 1) (x + 3)}{x + 3}$

Step 2.3.1.1.6.3

Move $3$ to the left of $A$ .

$x + 5 = A x + 3 \cdot A + \frac{(B) (x - 1) (x + 3)}{x + 3}$

Step 2.3.1.1.6.4

Cancel the common factor of $x + 3$ .

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Step 2.3.1.1.6.4.1

Cancel the common factor.

$x + 5 = A x + 3 A + \frac{(B) (x - 1) (x + 3)}{x + 3}$

Step 2.3.1.1.6.4.2

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

$x + 5 = A x + 3 A + (B) (x - 1)$

Step 2.3.1.1.6.5

Apply the distributive property.

$x + 5 = A x + 3 A + B x + B \cdot - 1$

Step 2.3.1.1.6.6

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

$x + 5 = A x + 3 A + B x - 1 \cdot B$

Step 2.3.1.1.6.7

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

$x + 5 = A x + 3 A + B x - B$

Step 2.3.1.1.7

Move $3 A$ .

$x + 5 = A x + B x + 3 A - B$

Step 2.3.1.2

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

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Step 2.3.1.2.1

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.

$1 = A + B$

Step 2.3.1.2.2

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.

$5 = 3 A - 1 B$

Step 2.3.1.2.3

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

$1 = A + B$

$5 = 3 A - 1 B$

$1 = A + B$

$5 = 3 A - 1 B$

Step 2.3.1.3

Solve the system of equations.

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Step 2.3.1.3.1

Solve for $A$ in $1 = A + B$ .

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Step 2.3.1.3.1.1

Rewrite the equation as $A + B = 1$ .

$A + B = 1$

$5 = 3 A - 1 B$

Step 2.3.1.3.1.2

Subtract $B$ from both sides of the equation.

$A = 1 - B$

$5 = 3 A - 1 B$

$A = 1 - B$

$5 = 3 A - 1 B$

Step 2.3.1.3.2

Replace all occurrences of $A$ with $1 - B$ in each equation.

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Step 2.3.1.3.2.1

Replace all occurrences of $A$ in $5 = 3 A - 1 B$ with $1 - B$ .

$5 = 3 (1 - B) - 1 B$

$A = 1 - B$

Step 2.3.1.3.2.2

Simplify the right side.

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Step 2.3.1.3.2.2.1

Simplify $3 (1 - B) - 1 B$ .

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Step 2.3.1.3.2.2.1.1

Simplify each term.

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Step 2.3.1.3.2.2.1.1.1

Apply the distributive property.

$5 = 3 \cdot 1 + 3 (- B) - 1 B$

$A = 1 - B$

Step 2.3.1.3.2.2.1.1.2

Multiply $3$ by $1$ .

$5 = 3 + 3 (- B) - 1 B$

$A = 1 - B$

Step 2.3.1.3.2.2.1.1.3

Multiply $- 1$ by $3$ .

$5 = 3 - 3 B - 1 B$

$A = 1 - B$

Step 2.3.1.3.2.2.1.1.4

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

$5 = 3 - 3 B - B$

$A = 1 - B$

$5 = 3 - 3 B - B$

$A = 1 - B$

Step 2.3.1.3.2.2.1.2

Subtract $B$ from $- 3 B$ .

$5 = 3 - 4 B$

$A = 1 - B$

$5 = 3 - 4 B$

$A = 1 - B$

$5 = 3 - 4 B$

$A = 1 - B$

$5 = 3 - 4 B$

$A = 1 - B$

Step 2.3.1.3.3

Solve for $B$ in $5 = 3 - 4 B$ .

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Step 2.3.1.3.3.1

Rewrite the equation as $3 - 4 B = 5$ .

$3 - 4 B = 5$

$A = 1 - B$

Step 2.3.1.3.3.2

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

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Step 2.3.1.3.3.2.1

Subtract $3$ from both sides of the equation.

$- 4 B = 5 - 3$

$A = 1 - B$

Step 2.3.1.3.3.2.2

Subtract $3$ from $5$ .

$- 4 B = 2$

$A = 1 - B$

$- 4 B = 2$

$A = 1 - B$

Step 2.3.1.3.3.3

Divide each term in $- 4 B = 2$ by $- 4$ and simplify.

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Step 2.3.1.3.3.3.1

Divide each term in $- 4 B = 2$ by $- 4$ .

$\frac{- 4 B}{- 4} = \frac{2}{- 4}$

$A = 1 - B$

Step 2.3.1.3.3.3.2

Simplify the left side.

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Step 2.3.1.3.3.3.2.1

Cancel the common factor of $- 4$ .

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Step 2.3.1.3.3.3.2.1.1

Cancel the common factor.

$\frac{- 4 B}{- 4} = \frac{2}{- 4}$

$A = 1 - B$

Step 2.3.1.3.3.3.2.1.2

Divide $B$ by $1$ .

$B = \frac{2}{- 4}$

$A = 1 - B$

$B = \frac{2}{- 4}$

$A = 1 - B$

$B = \frac{2}{- 4}$

$A = 1 - B$

Step 2.3.1.3.3.3.3

Simplify the right side.

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Step 2.3.1.3.3.3.3.1

Cancel the common factor of $2$ and $- 4$ .

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Step 2.3.1.3.3.3.3.1.1

Factor $2$ out of $2$ .

$B = \frac{2 (1)}{- 4}$

$A = 1 - B$

Step 2.3.1.3.3.3.3.1.2

Cancel the common factors.

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Step 2.3.1.3.3.3.3.1.2.1

Factor $2$ out of $- 4$ .

$B = \frac{2 \cdot 1}{2 \cdot - 2}$

$A = 1 - B$

Step 2.3.1.3.3.3.3.1.2.2

Cancel the common factor.

$B = \frac{2 \cdot 1}{2 \cdot - 2}$

$A = 1 - B$

Step 2.3.1.3.3.3.3.1.2.3

Rewrite the expression.

$B = \frac{1}{- 2}$

$A = 1 - B$

$B = \frac{1}{- 2}$

$A = 1 - B$

$B = \frac{1}{- 2}$

$A = 1 - B$

Step 2.3.1.3.3.3.3.2

Move the negative in front of the fraction.

$B = - \frac{1}{2}$

$A = 1 - B$

$B = - \frac{1}{2}$

$A = 1 - B$

$B = - \frac{1}{2}$

$A = 1 - B$

$B = - \frac{1}{2}$

$A = 1 - B$

Step 2.3.1.3.4

Replace all occurrences of $B$ with $- \frac{1}{2}$ in each equation.

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Step 2.3.1.3.4.1

Replace all occurrences of $B$ in $A = 1 - B$ with $- \frac{1}{2}$ .

$A = 1 - (- \frac{1}{2})$

$B = - \frac{1}{2}$

Step 2.3.1.3.4.2

Simplify the right side.

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Step 2.3.1.3.4.2.1

Simplify $1 - (- \frac{1}{2})$ .

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Step 2.3.1.3.4.2.1.1

Multiply $- (- \frac{1}{2})$ .

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Step 2.3.1.3.4.2.1.1.1

Multiply $- 1$ by $- 1$ .

$A = 1 + 1 (\frac{1}{2})$

$B = - \frac{1}{2}$

Step 2.3.1.3.4.2.1.1.2

Multiply $\frac{1}{2}$ by $1$ .

$A = 1 + \frac{1}{2}$

$B = - \frac{1}{2}$

$A = 1 + \frac{1}{2}$

$B = - \frac{1}{2}$

Step 2.3.1.3.4.2.1.2

Write $1$ as a fraction with a common denominator.

$A = \frac{2}{2} + \frac{1}{2}$

$B = - \frac{1}{2}$

Step 2.3.1.3.4.2.1.3

Combine the numerators over the common denominator.

$A = \frac{2 + 1}{2}$

$B = - \frac{1}{2}$

Step 2.3.1.3.4.2.1.4

Add $2$ and $1$ .

$A = \frac{3}{2}$

$B = - \frac{1}{2}$

$A = \frac{3}{2}$

$B = - \frac{1}{2}$

$A = \frac{3}{2}$

$B = - \frac{1}{2}$

$A = \frac{3}{2}$

$B = - \frac{1}{2}$

Step 2.3.1.3.5

List all of the solutions.

$A = \frac{3}{2}, B = - \frac{1}{2}$

Step 2.3.1.4

Replace each of the partial fraction coefficients in $\frac{A}{x - 1} + \frac{B}{x + 3}$ with the values found for $A$ and $B$ .

$\frac{\frac{3}{2}}{x - 1} + \frac{- \frac{1}{2}}{x + 3}$

Step 2.3.1.5

Simplify.

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Step 2.3.1.5.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{3}{2} \cdot \frac{1}{x - 1} + \frac{- \frac{1}{2}}{x + 3}$

Step 2.3.1.5.2

Multiply $\frac{3}{2}$ by $\frac{1}{x - 1}$ .

$\frac{3}{2 (x - 1)} + \frac{- \frac{1}{2}}{x + 3}$

Step 2.3.1.5.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{3}{2 (x - 1)} - \frac{1}{2} \cdot \frac{1}{x + 3}$

Step 2.3.1.5.4

Multiply $\frac{1}{x + 3}$ by $\frac{1}{2}$ .

$\frac{3}{2 (x - 1)} - \frac{1}{(x + 3) \cdot 2}$

Step 2.3.1.5.5

Move $2$ to the left of $x + 3$ .

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

Step 2.3.2

Split the single integral into multiple integrals.

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

Step 2.3.3

Since $\frac{3}{2}$ is constant with respect to $x$ , move $\frac{3}{2}$ out of the integral.

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

Step 2.3.4

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

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Step 2.3.4.1

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

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Step 2.3.4.1.1

Differentiate $x - 1$ .

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

Step 2.3.4.1.2

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 2.3.4.1.3

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 2.3.4.1.4

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

$1 + 0$

Step 2.3.4.1.5

Add $1$ and $0$ .

$1$

Step 2.3.4.2

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

$y + C_{1} = \frac{3}{2} \int \frac{1}{u_{1}} d u_{1} + \int - \frac{1}{2 (x + 3)} d x$

Step 2.3.5

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

$y + C_{1} = \frac{3}{2} (\ln (| u_{1} |) + C_{2}) + \int - \frac{1}{2 (x + 3)} d x$

Step 2.3.6

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

$y + C_{1} = \frac{3}{2} (\ln (| u_{1} |) + C_{2}) - \int \frac{1}{2 (x + 3)} d x$

Step 2.3.7

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

$y + C_{1} = \frac{3}{2} (\ln (| u_{1} |) + C_{2}) - (\frac{1}{2} \int \frac{1}{x + 3} d x)$

Step 2.3.8

Let $u_{2} = x + 3$ . Then $d u_{2} = d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 2.3.8.1

Let $u_{2} = x + 3$ . Find $\frac{d u_{2}}{d x}$ .

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Step 2.3.8.1.1

Differentiate $x + 3$ .

$\frac{d}{d x} [x + 3]$

Step 2.3.8.1.2

By the Sum Rule, the derivative of $x + 3$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [3]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [3]$

Step 2.3.8.1.3

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

$1 + \frac{d}{d x} [3]$

Step 2.3.8.1.4

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$1 + 0$

Step 2.3.8.1.5

Add $1$ and $0$ .

$1$

Step 2.3.8.2

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

$y + C_{1} = \frac{3}{2} (\ln (| u_{1} |) + C_{2}) - \frac{1}{2} \int \frac{1}{u_{2}} d u_{2}$

Step 2.3.9

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

$y + C_{1} = \frac{3}{2} (\ln (| u_{1} |) + C_{2}) - \frac{1}{2} (\ln (| u_{2} |) + C_{3})$

Step 2.3.10

Simplify.

$y + C_{1} = \frac{3}{2} \ln (| u_{1} |) - \frac{1}{2} \ln (| u_{2} |) + C_{4}$

Step 2.3.11

Substitute back in for each integration substitution variable.

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Step 2.3.11.1

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

$y + C_{1} = \frac{3}{2} \ln (| x - 1 |) - \frac{1}{2} \ln (| u_{2} |) + C_{4}$

Step 2.3.11.2

Replace all occurrences of $u_{2}$ with $x + 3$ .

$y + C_{1} = \frac{3}{2} \ln (| x - 1 |) - \frac{1}{2} \ln (| x + 3 |) + C_{4}$

Step 2.4

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

$y = \frac{3}{2} \ln (| x - 1 |) - \frac{1}{2} \ln (| x + 3 |) + C$