Calculus Examples

$\sin (2 x) + 3 \cos (3 x)$

Step 1

Write $\sin (2 x) + 3 \cos (3 x)$ as a function.

$f (x) = \sin (2 x) + 3 \cos (3 x)$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int \sin (2 x) + 3 \cos (3 x) d x$

Step 4

Split the single integral into multiple integrals.

$\int \sin (2 x) d x + \int 3 \cos (3 x) d x$

Step 5

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

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Step 5.1

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

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Step 5.1.1

Differentiate $2 x$ .

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

Step 5.1.2

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

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

Step 5.1.3

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

$2 \cdot 1$

Step 5.1.4

Multiply $2$ by $1$ .

$2$

Step 5.2

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

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

Step 10

Let $u_{2} = 3 x$ . Then $d u_{2} = 3 d x$ , so $\frac{1}{3} d u_{2} = d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 10.1

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

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Step 10.1.1

Differentiate $3 x$ .

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

Step 10.1.2

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 10.1.3

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 10.1.4

Multiply $3$ by $1$ .

$3$

Step 10.2

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

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

Step 11

Combine $\cos (u_{2})$ and $\frac{1}{3}$ .

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

Step 12

Since $\frac{1}{3}$ is constant with respect to $u_{2}$ , move $\frac{1}{3}$ out of the integral.

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

Step 13

Simplify.

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Step 13.1

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

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

Step 13.2

Cancel the common factor of $3$ .

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Step 13.2.1

Cancel the common factor.

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

Step 13.2.2

Rewrite the expression.

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

Step 13.3

Multiply $\int \cos (u_{2}) d u_{2}$ by $1$ .

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

Step 14

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

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

Step 15

Simplify.

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

Step 16

Substitute back in for each integration substitution variable.

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Step 16.1

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

$- \frac{\cos (2 x)}{2} + \sin (u_{2}) + C$

Step 16.2

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

$- \frac{\cos (2 x)}{2} + \sin (3 x) + C$

Step 17

Reorder terms.

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

Step 18

The answer is the antiderivative of the function $f (x) = \sin (2 x) + 3 \cos (3 x)$ .

$F (x) =$ $- \frac{1}{2} \cos (2 x) + \sin (3 x) + C$