Calculus Examples

$\int \sin (3 x) + e^{2 x + 1} d x$

Step 1

Split the single integral into multiple integrals.

$\int \sin (3 x) d x + \int e^{2 x + 1} d x$

Step 2

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

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Step 2.1

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

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Step 2.1.1

Differentiate $3 x$ .

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

Step 2.1.2

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

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

Step 2.1.3

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

$\int \sin (u_{1}) \frac{1}{3} d u_{1} + \int e^{2 x + 1} d x$

Step 3

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

$\int \frac{\sin (u_{1})}{3} d u_{1} + \int e^{2 x + 1} d x$

Step 4

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

$\frac{1}{3} \int \sin (u_{1}) d u_{1} + \int e^{2 x + 1} d x$

Step 5

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

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

Step 6

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

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Step 6.1

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

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Step 6.1.1

Differentiate $2 x + 1$ .

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

Step 6.1.2

By the Sum Rule, the derivative of $2 x + 1$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [1]$ .

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

Step 6.1.3

Evaluate $\frac{d}{d x} [2 x]$ .

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Step 6.1.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

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

Step 6.1.3.2

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

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

Step 6.1.3.3

Multiply $2$ by $1$ .

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

Step 6.1.4

Differentiate using the Constant Rule.

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Step 6.1.4.1

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

$2 + 0$

Step 6.1.4.2

Add $2$ and $0$ .

$2$

Step 6.2

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

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

Step 7

Combine $e^{u_{2}}$ and $\frac{1}{2}$ .

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

Step 8

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

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

Step 9

The integral of $e^{u_{2}}$ with respect to $u_{2}$ is $e^{u_{2}}$ .

$\frac{1}{3} (- \cos (u_{1}) + C) + \frac{1}{2} (e^{u_{2}} + C)$

Step 10

Simplify.

$- \frac{\cos (u_{1})}{3} + \frac{1}{2} e^{u_{2}} + C$

Step 11

Substitute back in for each integration substitution variable.

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Step 11.1

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

$- \frac{\cos (3 x)}{3} + \frac{1}{2} e^{u_{2}} + C$

Step 11.2

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

$- \frac{\cos (3 x)}{3} + \frac{1}{2} e^{2 x + 1} + C$

Step 12

Reorder terms.

$- \frac{1}{3} \cos (3 x) + \frac{1}{2} e^{2 x + 1} + C$