Calculus Examples

$\int \sin (3 y) + \cos (5 y) d y$

Step 1

Split the single integral into multiple integrals.

$\int \sin (3 y) d y + \int \cos (5 y) d y$

Step 2

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 $d$ $u_{1}$ .

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Step 2.1

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

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Step 2.1.1

Differentiate $3 y$ .

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

Step 2.1.2

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 2.1.3

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 2.1.4

Multiply $3$ by $1$ .

$3$

Step 2.2

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

$\int \sin (u_{1}) \frac{1}{3} d u_{1} + \int \cos (5 y) d y$

Step 3

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

$\int \frac{\sin (u_{1})}{3} d u_{1} + \int \cos (5 y) d y$

Step 4

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

$\frac{1}{3} \int \sin (u_{1}) d u_{1} + \int \cos (5 y) d y$

Step 5

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

$\frac{1}{3} (- \cos (u_{1}) + C) + \int \cos (5 y) d y$

Step 6

Let $u_{2} = 5 y$ . Then $d u_{2} = 5 d y$ , so $\frac{1}{5} d u_{2} = d y$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 6.1

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

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Step 6.1.1

Differentiate $5 y$ .

$\frac{d}{d y} [5 y]$

Step 6.1.2

Since $5$ is constant with respect to $y$ , the derivative of $5 y$ with respect to $y$ is $5 \frac{d}{d y} [y]$ .

$5 \frac{d}{d y} [y]$

Step 6.1.3

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

$5 \cdot 1$

Step 6.1.4

Multiply $5$ by $1$ .

$5$

Step 6.2

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

$\frac{1}{3} (- \cos (u_{1}) + C) + \int \cos (u_{2}) \frac{1}{5} d u_{2}$

Step 7

Combine $\cos (u_{2})$ and $\frac{1}{5}$ .

$\frac{1}{3} (- \cos (u_{1}) + C) + \int \frac{\cos (u_{2})}{5} d u_{2}$

Step 8

Since $\frac{1}{5}$ is constant with respect to $u_{2}$ , move $\frac{1}{5}$ out of the integral.

$\frac{1}{3} (- \cos (u_{1}) + C) + \frac{1}{5} \int \cos (u_{2}) d u_{2}$

Step 9

The integral of $\cos (u_{2})$ with respect to $u_{2}$ is $\sin (u_{2})$ .

$\frac{1}{3} (- \cos (u_{1}) + C) + \frac{1}{5} (\sin (u_{2}) + C)$

Step 10

Simplify.

$- \frac{\cos (u_{1})}{3} + \frac{1}{5} \sin (u_{2}) + C$

Step 11

Substitute back in for each integration substitution variable.

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Step 11.1

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

$- \frac{\cos (3 y)}{3} + \frac{1}{5} \sin (u_{2}) + C$

Step 11.2

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

$- \frac{\cos (3 y)}{3} + \frac{1}{5} \sin (5 y) + C$

Step 12

Reorder terms.

$- \frac{1}{3} \cos (3 y) + \frac{1}{5} \sin (5 y) + C$