Calculus Examples

$(1 - x^{2}) \cdot \cos (3 y) d y - 7 d x = 0$

Step 1

Add $7 d x$ to both sides of the equation.

$(1 - x^{2}) \cdot \cos (3 y) d y = 7 d x$

Step 2

Multiply both sides by $\frac{1}{1 - x^{2}}$ .

$\frac{1}{1 - x^{2}} ((1 - x^{2}) \cdot \cos (3 y)) d y = \frac{1}{1 - x^{2}} \cdot 7 d x$

Step 3

Simplify.

Tap for more steps...

Step 3.1

Cancel the common factor of $1 - x^{2}$ .

Tap for more steps...

Step 3.1.1

Factor $1 - x^{2}$ out of $(1 - x^{2}) \cdot \cos (3 y)$ .

$\frac{1}{1 - x^{2}} ((1 - x^{2}) (\cos (3 y))) d y = \frac{1}{1 - x^{2}} \cdot 7 d x$

Step 3.1.2

Cancel the common factor.

$\frac{1}{1 - x^{2}} ((1 - x^{2}) \cos (3 y)) d y = \frac{1}{1 - x^{2}} \cdot 7 d x$

Step 3.1.3

Rewrite the expression.

$\cos (3 y) d y = \frac{1}{1 - x^{2}} \cdot 7 d x$

Step 3.2

Simplify the denominator.

Tap for more steps...

Step 3.2.1

Rewrite $1$ as $1^{2}$ .

$\cos (3 y) d y = \frac{1}{1^{2} - x^{2}} \cdot 7 d x$

Step 3.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = x$ .

$\cos (3 y) d y = \frac{1}{(1 + x) (1 - x)} \cdot 7 d x$

Step 3.3

Combine $\frac{1}{(1 + x) (1 - x)}$ and $7$ .

$\cos (3 y) d y = \frac{7}{(1 + x) (1 - x)} d x$

Step 4

Integrate both sides.

Tap for more steps...

Step 4.1

Set up an integral on each side.

$\int \cos (3 y) d y = \int \frac{7}{(1 + x) (1 - x)} d x$

Step 4.2

Integrate the left side.

Tap for more steps...

Step 4.2.1

Let $u_{1} = 3 y$ . Then $d u_{1} = 3 d y$ , so $\frac{1}{3} d u_{1} = d y$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

Tap for more steps...

Step 4.2.1.1

Let $u_{1} = 3 y$ . Find $\frac{d u_{1}}{d y}$ .

Tap for more steps...

Step 4.2.1.1.1

Differentiate $3 y$ .

$\frac{d}{d y} [3 y]$

Step 4.2.1.1.2

Since $3$ is constant with respect to $y$ , the derivative of $3 y$ with respect to $y$ is $3 \frac{d}{d y} [y]$ .

$3 \frac{d}{d y} [y]$

Step 4.2.1.1.3

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

$3 \cdot 1$

Step 4.2.1.1.4

Multiply $3$ by $1$ .

$3$

Step 4.2.1.2

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

$\int \cos (u_{1}) \frac{1}{3} d u_{1} = \int \frac{7}{(1 + x) (1 - x)} d x$

Step 4.2.2

Combine $\cos (u_{1})$ and $\frac{1}{3}$ .

$\int \frac{\cos (u_{1})}{3} d u_{1} = \int \frac{7}{(1 + x) (1 - x)} d x$

Step 4.2.3

Since $\frac{1}{3}$ is constant with respect to $u_{1}$ , move $\frac{1}{3}$ out of the integral.

$\frac{1}{3} \int \cos (u_{1}) d u_{1} = \int \frac{7}{(1 + x) (1 - x)} d x$

Step 4.2.4

The integral of $\cos (u_{1})$ with respect to $u_{1}$ is $\sin (u_{1})$ .

$\frac{1}{3} (\sin (u_{1}) + C_{1}) = \int \frac{7}{(1 + x) (1 - x)} d x$

Step 4.2.5

Simplify.

$\frac{1}{3} \sin (u_{1}) + C_{1} = \int \frac{7}{(1 + x) (1 - x)} d x$

Step 4.2.6

Replace all occurrences of $u_{1}$ with $3 y$ .

$\frac{1}{3} \sin (3 y) + C_{1} = \int \frac{7}{(1 + x) (1 - x)} d x$

Step 4.3

Integrate the right side.

Tap for more steps...

Step 4.3.1

Since $7$ is constant with respect to $x$ , move $7$ out of the integral.

$\frac{1}{3} \sin (3 y) + C_{1} = 7 \int \frac{1}{(1 + x) (1 - x)} d x$

Step 4.3.2

Write the fraction using partial fraction decomposition.

Tap for more steps...

Step 4.3.2.1

Decompose the fraction and multiply through by the common denominator.

Tap for more steps...

Step 4.3.2.1.1

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 $A$ .

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

Step 4.3.2.1.2

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

Step 4.3.2.1.3

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

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

Step 4.3.2.1.4

Cancel the common factor of $1 + x$ .

Tap for more steps...

Step 4.3.2.1.4.1

Cancel the common factor.

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

Step 4.3.2.1.4.2

Rewrite the expression.

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

Step 4.3.2.1.5

Cancel the common factor of $1 - x$ .

Tap for more steps...

Step 4.3.2.1.5.1

Cancel the common factor.

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

Step 4.3.2.1.5.2

Rewrite the expression.

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

Step 4.3.2.1.6

Simplify each term.

Tap for more steps...

Step 4.3.2.1.6.1

Cancel the common factor of $1 + x$ .

Tap for more steps...

Step 4.3.2.1.6.1.1

Cancel the common factor.

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

Step 4.3.2.1.6.1.2

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

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

Step 4.3.2.1.6.2

Apply the distributive property.

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

Step 4.3.2.1.6.3

Multiply $A$ by $1$ .

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

Step 4.3.2.1.6.4

Rewrite using the commutative property of multiplication.

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

Step 4.3.2.1.6.5

Cancel the common factor of $1 - x$ .

Tap for more steps...

Step 4.3.2.1.6.5.1

Cancel the common factor.

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

Step 4.3.2.1.6.5.2

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

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

Step 4.3.2.1.6.6

Apply the distributive property.

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

Step 4.3.2.1.6.7

Multiply $B$ by $1$ .

$1 = A - A x + B + B x$

Step 4.3.2.1.7

Simplify the expression.

Tap for more steps...

Step 4.3.2.1.7.1

Move $A$ .

$1 = A - x A + B + B x$

Step 4.3.2.1.7.2

Reorder $B$ and $x$ .

$1 = A - x A + B + B x$

Step 4.3.2.1.7.3

Move $B$ .

$1 = A - x A + B x + B$

Step 4.3.2.1.7.4

Move $A$ .

$1 = - x A + B x + A + B$

Step 4.3.2.2

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

Tap for more steps...

Step 4.3.2.2.1

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.

$0 = - 1 A + B$

Step 4.3.2.2.2

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.

$1 = A + B$

Step 4.3.2.2.3

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

$0 = - 1 A + B$

$1 = A + B$

$0 = - 1 A + B$

$1 = A + B$

Step 4.3.2.3

Solve the system of equations.

Tap for more steps...

Step 4.3.2.3.1

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

Tap for more steps...

Step 4.3.2.3.1.1

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

$A + B = 1$

$0 = - 1 A + B$

Step 4.3.2.3.1.2

Subtract $B$ from both sides of the equation.

$A = 1 - B$

$0 = - 1 A + B$

$A = 1 - B$

$0 = - 1 A + B$

Step 4.3.2.3.2

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

Tap for more steps...

Step 4.3.2.3.2.1

Replace all occurrences of $A$ in $0 = - 1 A + B$ with $1 - B$ .

$0 = - 1 (1 - B) + B$

$A = 1 - B$

Step 4.3.2.3.2.2

Simplify the right side.

Tap for more steps...

Step 4.3.2.3.2.2.1

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

Tap for more steps...

Step 4.3.2.3.2.2.1.1

Simplify each term.

Tap for more steps...

Step 4.3.2.3.2.2.1.1.1

Apply the distributive property.

$0 = - 1 \cdot 1 - 1 (- B) + B$

$A = 1 - B$

Step 4.3.2.3.2.2.1.1.2

Multiply $- 1$ by $1$ .

$0 = - 1 - 1 (- B) + B$

$A = 1 - B$

Step 4.3.2.3.2.2.1.1.3

Multiply $- 1 (- B)$ .

Tap for more steps...

Step 4.3.2.3.2.2.1.1.3.1

Multiply $- 1$ by $- 1$ .

$0 = - 1 + 1 B + B$

$A = 1 - B$

Step 4.3.2.3.2.2.1.1.3.2

Multiply $B$ by $1$ .

$0 = - 1 + B + B$

$A = 1 - B$

$0 = - 1 + B + B$

$A = 1 - B$

$0 = - 1 + B + B$

$A = 1 - B$

Step 4.3.2.3.2.2.1.2

Add $B$ and $B$ .

$0 = - 1 + 2 B$

$A = 1 - B$

$0 = - 1 + 2 B$

$A = 1 - B$

$0 = - 1 + 2 B$

$A = 1 - B$

$0 = - 1 + 2 B$

$A = 1 - B$

Step 4.3.2.3.3

Solve for $B$ in $0 = - 1 + 2 B$ .

Tap for more steps...

Step 4.3.2.3.3.1

Rewrite the equation as $- 1 + 2 B = 0$ .

$- 1 + 2 B = 0$

$A = 1 - B$

Step 4.3.2.3.3.2

Add $1$ to both sides of the equation.

$2 B = 1$

$A = 1 - B$

Step 4.3.2.3.3.3

Divide each term in $2 B = 1$ by $2$ and simplify.

Tap for more steps...

Step 4.3.2.3.3.3.1

Divide each term in $2 B = 1$ by $2$ .

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

$A = 1 - B$

Step 4.3.2.3.3.3.2

Simplify the left side.

Tap for more steps...

Step 4.3.2.3.3.3.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.3.2.3.3.3.2.1.1

Cancel the common factor.

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

$A = 1 - B$

Step 4.3.2.3.3.3.2.1.2

Divide $B$ by $1$ .

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

$A = 1 - B$

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

$A = 1 - B$

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

$A = 1 - B$

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

$A = 1 - B$

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

$A = 1 - B$

Step 4.3.2.3.4

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

Tap for more steps...

Step 4.3.2.3.4.1

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

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

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

Step 4.3.2.3.4.2

Simplify the right side.

Tap for more steps...

Step 4.3.2.3.4.2.1

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

Tap for more steps...

Step 4.3.2.3.4.2.1.1

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

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

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

Step 4.3.2.3.4.2.1.2

Combine the numerators over the common denominator.

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

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

Step 4.3.2.3.4.2.1.3

Subtract $1$ from $2$ .

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

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

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

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

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

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

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

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

Step 4.3.2.3.5

List all of the solutions.

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

Step 4.3.2.4

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

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

Step 4.3.2.5

Simplify.

Tap for more steps...

Step 4.3.2.5.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.3.2.5.2

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

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

Step 4.3.2.5.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.3.2.5.4

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

$\frac{1}{3} \sin (3 y) + C_{1} = 7 \int \frac{1}{2 (1 + x)} + \frac{1}{2 (1 - x)} d x$

Step 4.3.3

Split the single integral into multiple integrals.

$\frac{1}{3} \sin (3 y) + C_{1} = 7 (\int \frac{1}{2 (1 + x)} d x + \int \frac{1}{2 (1 - x)} d x)$

Step 4.3.4

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

$\frac{1}{3} \sin (3 y) + C_{1} = 7 (\frac{1}{2} \int \frac{1}{1 + x} d x + \int \frac{1}{2 (1 - x)} d x)$

Step 4.3.5

Let $u_{2} = 1 + x$ . Then $d u_{2} = d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

Tap for more steps...

Step 4.3.5.1

Let $u_{2} = 1 + x$ . Find $\frac{d u_{2}}{d x}$ .

Tap for more steps...

Step 4.3.5.1.1

Differentiate $1 + x$ .

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

Step 4.3.5.1.2

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

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

Step 4.3.5.1.3

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

$0 + \frac{d}{d x} [x]$

Step 4.3.5.1.4

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

$0 + 1$

Step 4.3.5.1.5

Add $0$ and $1$ .

$1$

Step 4.3.5.2

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

$\frac{1}{3} \sin (3 y) + C_{1} = 7 (\frac{1}{2} \int \frac{1}{u_{2}} d u_{2} + \int \frac{1}{2 (1 - x)} d x)$

Step 4.3.6

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

$\frac{1}{3} \sin (3 y) + C_{1} = 7 (\frac{1}{2} (\ln (| u_{2} |) + C_{2}) + \int \frac{1}{2 (1 - x)} d x)$

Step 4.3.7

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

$\frac{1}{3} \sin (3 y) + C_{1} = 7 (\frac{1}{2} (\ln (| u_{2} |) + C_{2}) + \frac{1}{2} \int \frac{1}{1 - x} d x)$

Step 4.3.8

Let $u_{3} = 1 - x$ . Then $d u_{3} = - d x$ , so $- d u_{3} = d x$ . Rewrite using $u_{3}$ and $d$ $u_{3}$ .

Tap for more steps...

Step 4.3.8.1

Let $u_{3} = 1 - x$ . Find $\frac{d u_{3}}{d x}$ .

Tap for more steps...

Step 4.3.8.1.1

Rewrite.

$\frac{- 1}{1}$

Step 4.3.8.1.2

Divide $- 1$ by $1$ .

$- 1$

Step 4.3.8.2

Rewrite the problem using $u_{3}$ and $d u_{3}$ .

$\frac{1}{3} \sin (3 y) + C_{1} = 7 (\frac{1}{2} (\ln (| u_{2} |) + C_{2}) + \frac{1}{2} \int \frac{- 1}{u_{3}} d u_{3})$

Step 4.3.9

Move the negative in front of the fraction.

$\frac{1}{3} \sin (3 y) + C_{1} = 7 (\frac{1}{2} (\ln (| u_{2} |) + C_{2}) + \frac{1}{2} \int - \frac{1}{u_{3}} d u_{3})$

Step 4.3.10

Since $- 1$ is constant with respect to $u_{3}$ , move $- 1$ out of the integral.

$\frac{1}{3} \sin (3 y) + C_{1} = 7 (\frac{1}{2} (\ln (| u_{2} |) + C_{2}) + \frac{1}{2} (- \int \frac{1}{u_{3}} d u_{3}))$

Step 4.3.11

The integral of $\frac{1}{u_{3}}$ with respect to $u_{3}$ is $\ln (| u_{3} |)$ .

$\frac{1}{3} \sin (3 y) + C_{1} = 7 (\frac{1}{2} (\ln (| u_{2} |) + C_{2}) + \frac{1}{2} (- (\ln (| u_{3} |) + C_{3})))$

Step 4.3.12

Simplify.

$\frac{1}{3} \sin (3 y) + C_{1} = 7 (\frac{\ln (| u_{2} |)}{2} - \frac{\ln (| u_{3} |)}{2}) + C_{4}$

Step 4.3.13

Substitute back in for each integration substitution variable.

Tap for more steps...

Step 4.3.13.1

Replace all occurrences of $u_{2}$ with $1 + x$ .

$\frac{1}{3} \sin (3 y) + C_{1} = 7 (\frac{\ln (| 1 + x |)}{2} - \frac{\ln (| u_{3} |)}{2}) + C_{4}$

Step 4.3.13.2

Replace all occurrences of $u_{3}$ with $1 - x$ .

$\frac{1}{3} \sin (3 y) + C_{1} = 7 (\frac{\ln (| 1 + x |)}{2} - \frac{\ln (| 1 - x |)}{2}) + C_{4}$

Step 4.3.14

Reorder terms.

$\frac{1}{3} \sin (3 y) + C_{1} = 7 (\frac{1}{2} \ln (| 1 + x |) - \frac{1}{2} \ln (| 1 - x |)) + C_{4}$

Step 4.4

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

$\frac{1}{3} \sin (3 y) = 7 (\frac{1}{2} \ln (| 1 + x |) - \frac{1}{2} \ln (| 1 - x |)) + C$

Step 5

Solve for $y$ .

Tap for more steps...

Step 5.1

Simplify the expressions in the equation.

Tap for more steps...

Step 5.1.1

Simplify the left side.

Tap for more steps...

Step 5.1.1.1

Combine $\frac{1}{3}$ and $\sin (3 y)$ .

$\frac{\sin (3 y)}{3} = 7 (\frac{1}{2} \ln (| 1 + x |) - \frac{1}{2} \ln (| 1 - x |)) + C$

Step 5.1.2

Simplify the right side.

Tap for more steps...

Step 5.1.2.1

Simplify each term.

Tap for more steps...

Step 5.1.2.1.1

Simplify each term.

Tap for more steps...

Step 5.1.2.1.1.1

Combine $\frac{1}{2}$ and $\ln (| 1 + x |)$ .

$\frac{\sin (3 y)}{3} = 7 (\frac{\ln (| 1 + x |)}{2} - \frac{1}{2} \ln (| 1 - x |)) + C$

Step 5.1.2.1.1.2

Combine $\ln (| 1 - x |)$ and $\frac{1}{2}$ .

$\frac{\sin (3 y)}{3} = 7 (\frac{\ln (| 1 + x |)}{2} - \frac{\ln (| 1 - x |)}{2}) + C$

Step 5.1.2.1.2

Apply the distributive property.

$\frac{\sin (3 y)}{3} = 7 \frac{\ln (| 1 + x |)}{2} + 7 (- \frac{\ln (| 1 - x |)}{2}) + C$

Step 5.1.2.1.3

Combine $7$ and $\frac{\ln (| 1 + x |)}{2}$ .

$\frac{\sin (3 y)}{3} = \frac{7 \ln (| 1 + x |)}{2} + 7 (- \frac{\ln (| 1 - x |)}{2}) + C$

Step 5.1.2.1.4

Multiply $7 (- \frac{\ln (| 1 - x |)}{2})$ .

Tap for more steps...

Step 5.1.2.1.4.1

Multiply $- 1$ by $7$ .

$\frac{\sin (3 y)}{3} = \frac{7 \ln (| 1 + x |)}{2} - 7 \frac{\ln (| 1 - x |)}{2} + C$

Step 5.1.2.1.4.2

Combine $- 7$ and $\frac{\ln (| 1 - x |)}{2}$ .

$\frac{\sin (3 y)}{3} = \frac{7 \ln (| 1 + x |)}{2} + \frac{- 7 \ln (| 1 - x |)}{2} + C$

Step 5.1.2.1.5

Move the negative in front of the fraction.

$\frac{\sin (3 y)}{3} = \frac{7 \ln (| 1 + x |)}{2} - \frac{7 \ln (| 1 - x |)}{2} + C$

Step 5.2

Multiply each term in $\frac{\sin (3 y)}{3} = \frac{7 \ln (| 1 + x |)}{2} - \frac{7 \ln (| 1 - x |)}{2} + C$ by $6$ to eliminate the fractions.

Tap for more steps...

Step 5.2.1

Multiply each term in $\frac{\sin (3 y)}{3} = \frac{7 \ln (| 1 + x |)}{2} - \frac{7 \ln (| 1 - x |)}{2} + C$ by $6$ .

$\frac{\sin (3 y)}{3} \cdot 6 = \frac{7 \ln (| 1 + x |)}{2} \cdot 6 - \frac{7 \ln (| 1 - x |)}{2} \cdot 6 + C \cdot 6$

Step 5.2.2

Simplify the left side.

Tap for more steps...

Step 5.2.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 5.2.2.1.1

Factor $3$ out of $6$ .

$\frac{\sin (3 y)}{3} \cdot (3 (2)) = \frac{7 \ln (| 1 + x |)}{2} \cdot 6 - \frac{7 \ln (| 1 - x |)}{2} \cdot 6 + C \cdot 6$

Step 5.2.2.1.2

Cancel the common factor.

$\frac{\sin (3 y)}{3} \cdot (3 \cdot 2) = \frac{7 \ln (| 1 + x |)}{2} \cdot 6 - \frac{7 \ln (| 1 - x |)}{2} \cdot 6 + C \cdot 6$

Step 5.2.2.1.3

Rewrite the expression.

$\sin (3 y) \cdot 2 = \frac{7 \ln (| 1 + x |)}{2} \cdot 6 - \frac{7 \ln (| 1 - x |)}{2} \cdot 6 + C \cdot 6$

Step 5.2.2.2

Move $2$ to the left of $\sin (3 y)$ .

$2 \sin (3 y) = \frac{7 \ln (| 1 + x |)}{2} \cdot 6 - \frac{7 \ln (| 1 - x |)}{2} \cdot 6 + C \cdot 6$

Step 5.2.3

Simplify the right side.

Tap for more steps...

Step 5.2.3.1

Simplify each term.

Tap for more steps...

Step 5.2.3.1.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.2.3.1.1.1

Factor $2$ out of $6$ .

$2 \sin (3 y) = \frac{7 \ln (| 1 + x |)}{2} \cdot (2 (3)) - \frac{7 \ln (| 1 - x |)}{2} \cdot 6 + C \cdot 6$

Step 5.2.3.1.1.2

Cancel the common factor.

$2 \sin (3 y) = \frac{7 \ln (| 1 + x |)}{2} \cdot (2 \cdot 3) - \frac{7 \ln (| 1 - x |)}{2} \cdot 6 + C \cdot 6$

Step 5.2.3.1.1.3

Rewrite the expression.

$2 \sin (3 y) = 7 \ln (| 1 + x |) \cdot 3 - \frac{7 \ln (| 1 - x |)}{2} \cdot 6 + C \cdot 6$

Step 5.2.3.1.2

Multiply $3$ by $7$ .

$2 \sin (3 y) = 21 \ln (| 1 + x |) - \frac{7 \ln (| 1 - x |)}{2} \cdot 6 + C \cdot 6$

Step 5.2.3.1.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.2.3.1.3.1

Move the leading negative in $- \frac{7 \ln (| 1 - x |)}{2}$ into the numerator.

$2 \sin (3 y) = 21 \ln (| 1 + x |) + \frac{- 7 \ln (| 1 - x |)}{2} \cdot 6 + C \cdot 6$

Step 5.2.3.1.3.2

Factor $2$ out of $6$ .

$2 \sin (3 y) = 21 \ln (| 1 + x |) + \frac{- 7 \ln (| 1 - x |)}{2} \cdot (2 (3)) + C \cdot 6$

Step 5.2.3.1.3.3

Cancel the common factor.

$2 \sin (3 y) = 21 \ln (| 1 + x |) + \frac{- 7 \ln (| 1 - x |)}{2} \cdot (2 \cdot 3) + C \cdot 6$

Step 5.2.3.1.3.4

Rewrite the expression.

$2 \sin (3 y) = 21 \ln (| 1 + x |) - 7 \ln (| 1 - x |) \cdot 3 + C \cdot 6$

Step 5.2.3.1.4

Multiply $3$ by $- 7$ .

$2 \sin (3 y) = 21 \ln (| 1 + x |) - 21 \ln (| 1 - x |) + C \cdot 6$

Step 5.2.3.1.5

Move $6$ to the left of $C$ .

$2 \sin (3 y) = 21 \ln (| 1 + x |) - 21 \ln (| 1 - x |) + 6 C$

Step 5.3

Simplify the right side.

Tap for more steps...

Step 5.3.1

Simplify $21 \ln (| 1 + x |) - 21 \ln (| 1 - x |) + 6 C$ .

Tap for more steps...

Step 5.3.1.1

Simplify each term.

Tap for more steps...

Step 5.3.1.1.1

Simplify $21 \ln (| 1 + x |)$ by moving $21$ inside the logarithm.

$2 \sin (3 y) = \ln ({| 1 + x |}^{21}) - 21 \ln (| 1 - x |) + 6 C$

Step 5.3.1.1.2

Simplify $- 21 \ln (| 1 - x |)$ by moving $21$ inside the logarithm.

$2 \sin (3 y) = \ln ({| 1 + x |}^{21}) - \ln ({| 1 - x |}^{21}) + 6 C$

Step 5.3.1.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$2 \sin (3 y) = \ln (\frac{{| 1 + x |}^{21}}{{| 1 - x |}^{21}}) + 6 C$

Step 5.4

Move all the terms containing a logarithm to the left side of the equation.

$- \ln (\frac{{| 1 + x |}^{21}}{{| 1 - x |}^{21}}) = - 2 \sin (3 y) + 6 C$

Step 5.5

Since $y$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$- 2 \sin (3 y) + 6 C = - \ln (\frac{{| 1 + x |}^{21}}{{| 1 - x |}^{21}})$

Step 5.6

Subtract $6 C$ from both sides of the equation.

$- 2 \sin (3 y) = - \ln (\frac{{| 1 + x |}^{21}}{{| 1 - x |}^{21}}) - 6 C$

Step 5.7

Divide each term in $- 2 \sin (3 y) = - \ln (\frac{{| 1 + x |}^{21}}{{| 1 - x |}^{21}}) - 6 C$ by $- 2$ and simplify.

Tap for more steps...

Step 5.7.1

Divide each term in $- 2 \sin (3 y) = - \ln (\frac{{| 1 + x |}^{21}}{{| 1 - x |}^{21}}) - 6 C$ by $- 2$ .

$\frac{- 2 \sin (3 y)}{- 2} = \frac{- \ln (\frac{{| 1 + x |}^{21}}{{| 1 - x |}^{21}})}{- 2} + \frac{- 6 C}{- 2}$

Step 5.7.2

Simplify the left side.

Tap for more steps...

Step 5.7.2.1

Cancel the common factor of $- 2$ .

Tap for more steps...

Step 5.7.2.1.1

Cancel the common factor.

$\frac{- 2 \sin (3 y)}{- 2} = \frac{- \ln (\frac{{| 1 + x |}^{21}}{{| 1 - x |}^{21}})}{- 2} + \frac{- 6 C}{- 2}$

Step 5.7.2.1.2

Divide $\sin (3 y)$ by $1$ .

$\sin (3 y) = \frac{- \ln (\frac{{| 1 + x |}^{21}}{{| 1 - x |}^{21}})}{- 2} + \frac{- 6 C}{- 2}$

Step 5.7.3

Simplify the right side.

Tap for more steps...

Step 5.7.3.1

Simplify each term.

Tap for more steps...

Step 5.7.3.1.1

Dividing two negative values results in a positive value.

$\sin (3 y) = \frac{\ln (\frac{{| 1 + x |}^{21}}{{| 1 - x |}^{21}})}{2} + \frac{- 6 C}{- 2}$

Step 5.7.3.1.2

Cancel the common factor of $- 6$ and $- 2$ .

Tap for more steps...

Step 5.7.3.1.2.1

Factor $- 2$ out of $- 6 C$ .

$\sin (3 y) = \frac{\ln (\frac{{| 1 + x |}^{21}}{{| 1 - x |}^{21}})}{2} + \frac{- 2 (3 C)}{- 2}$

Step 5.7.3.1.2.2

Cancel the common factors.

Tap for more steps...

Step 5.7.3.1.2.2.1

Factor $- 2$ out of $- 2$ .

$\sin (3 y) = \frac{\ln (\frac{{| 1 + x |}^{21}}{{| 1 - x |}^{21}})}{2} + \frac{- 2 (3 C)}{- 2 (1)}$

Step 5.7.3.1.2.2.2

Cancel the common factor.

$\sin (3 y) = \frac{\ln (\frac{{| 1 + x |}^{21}}{{| 1 - x |}^{21}})}{2} + \frac{- 2 (3 C)}{- 2 \cdot 1}$

Step 5.7.3.1.2.2.3

Rewrite the expression.

$\sin (3 y) = \frac{\ln (\frac{{| 1 + x |}^{21}}{{| 1 - x |}^{21}})}{2} + \frac{3 C}{1}$

Step 5.7.3.1.2.2.4

Divide $3 C$ by $1$ .

$\sin (3 y) = \frac{\ln (\frac{{| 1 + x |}^{21}}{{| 1 - x |}^{21}})}{2} + 3 C$

Step 5.8

Take the inverse sine of both sides of the equation to extract $y$ from inside the sine.

$3 y = \arcsin (\frac{\ln (\frac{{| 1 + x |}^{21}}{{| 1 - x |}^{21}})}{2} + 3 C)$

Step 5.9

Simplify the right side.

Tap for more steps...

Step 5.9.1

Simplify each term.

Tap for more steps...

Step 5.9.1.1

Rewrite $\frac{\ln (\frac{{| 1 + x |}^{21}}{{| 1 - x |}^{21}})}{2}$ as $\frac{1}{2} \ln (\frac{{| 1 + x |}^{21}}{{| 1 - x |}^{21}})$ .

$3 y = \arcsin (\frac{1}{2} \ln (\frac{{| 1 + x |}^{21}}{{| 1 - x |}^{21}}) + 3 C)$

Step 5.9.1.2

Simplify $\frac{1}{2} \ln (\frac{{| 1 + x |}^{21}}{{| 1 - x |}^{21}})$ by moving $\frac{1}{2}$ inside the logarithm.

$3 y = \arcsin (\ln ({(\frac{{| 1 + x |}^{21}}{{| 1 - x |}^{21}})}^{\frac{1}{2}}) + 3 C)$

Step 5.9.1.3

Apply the product rule to $\frac{{| 1 + x |}^{21}}{{| 1 - x |}^{21}}$ .

$3 y = \arcsin (\ln (\frac{{({| 1 + x |}^{21})}^{\frac{1}{2}}}{{({| 1 - x |}^{21})}^{\frac{1}{2}}}) + 3 C)$

Step 5.9.1.4

Multiply the exponents in ${({| 1 + x |}^{21})}^{\frac{1}{2}}$ .

Tap for more steps...

Step 5.9.1.4.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$3 y = \arcsin (\ln (\frac{{| 1 + x |}^{21 (\frac{1}{2})}}{{({| 1 - x |}^{21})}^{\frac{1}{2}}}) + 3 C)$

Step 5.9.1.4.2

Combine $21$ and $\frac{1}{2}$ .

$3 y = \arcsin (\ln (\frac{{| 1 + x |}^{\frac{21}{2}}}{{({| 1 - x |}^{21})}^{\frac{1}{2}}}) + 3 C)$

Step 5.9.1.5

Multiply the exponents in ${({| 1 - x |}^{21})}^{\frac{1}{2}}$ .

Tap for more steps...

Step 5.9.1.5.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$3 y = \arcsin (\ln (\frac{{| 1 + x |}^{\frac{21}{2}}}{{| 1 - x |}^{21 (\frac{1}{2})}}) + 3 C)$

Step 5.9.1.5.2

Combine $21$ and $\frac{1}{2}$ .

$3 y = \arcsin (\ln (\frac{{| 1 + x |}^{\frac{21}{2}}}{{| 1 - x |}^{\frac{21}{2}}}) + 3 C)$

Step 5.10

Divide each term in $3 y = \arcsin (\ln (\frac{{| 1 + x |}^{\frac{21}{2}}}{{| 1 - x |}^{\frac{21}{2}}}) + 3 C)$ by $3$ and simplify.

Tap for more steps...

Step 5.10.1

Divide each term in $3 y = \arcsin (\ln (\frac{{| 1 + x |}^{\frac{21}{2}}}{{| 1 - x |}^{\frac{21}{2}}}) + 3 C)$ by $3$ .

$\frac{3 y}{3} = \frac{\arcsin (\ln (\frac{{| 1 + x |}^{\frac{21}{2}}}{{| 1 - x |}^{\frac{21}{2}}}) + 3 C)}{3}$

Step 5.10.2

Simplify the left side.

Tap for more steps...

Step 5.10.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 5.10.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{\arcsin (\ln (\frac{{| 1 + x |}^{\frac{21}{2}}}{{| 1 - x |}^{\frac{21}{2}}}) + 3 C)}{3}$

Step 5.10.2.1.2

Divide $y$ by $1$ .

$y = \frac{\arcsin (\ln (\frac{{| 1 + x |}^{\frac{21}{2}}}{{| 1 - x |}^{\frac{21}{2}}}) + 3 C)}{3}$

Step 6

Simplify the constant of integration.

$y = \frac{\arcsin (\ln (\frac{{| 1 + x |}^{\frac{21}{2}}}{{| 1 - x |}^{\frac{21}{2}}}) + C)}{3}$