Calculus Examples

$\sin (3 x + \frac{π}{6})$

Step 1

Write $\sin (3 x + \frac{π}{6})$ as a function.

$f (x) = \sin (3 x + \frac{π}{6})$

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.

$F (x) = \int \sin (3 x + \frac{π}{6}) d x$

Step 4

Let $u = 3 x + \frac{π}{6}$ . Then $d u = 3 d x$ , so $\frac{1}{3} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.1

Let $u = 3 x + \frac{π}{6}$ . Find $\frac{d u}{d x}$ .

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Step 4.1.1

Differentiate $3 x + \frac{π}{6}$ .

$\frac{d}{d x} [3 x + \frac{π}{6}]$

Step 4.1.2

By the Sum Rule, the derivative of $3 x + \frac{π}{6}$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [\frac{π}{6}]$ .

$\frac{d}{d x} [3 x] + \frac{d}{d x} [\frac{π}{6}]$

Step 4.1.3

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

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Step 4.1.3.1

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

$3 \frac{d}{d x} [x] + \frac{d}{d x} [\frac{π}{6}]$

Step 4.1.3.2

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

$3 \cdot 1 + \frac{d}{d x} [\frac{π}{6}]$

Step 4.1.3.3

Multiply $3$ by $1$ .

$3 + \frac{d}{d x} [\frac{π}{6}]$

Step 4.1.4

Differentiate using the Constant Rule.

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Step 4.1.4.1

Since $\frac{π}{6}$ is constant with respect to $x$ , the derivative of $\frac{π}{6}$ with respect to $x$ is $0$ .

$3 + 0$

Step 4.1.4.2

Add $3$ and $0$ .

$3$

Step 4.2

Rewrite the problem using $u$ and $d u$ .

$\int \sin (u) \frac{1}{3} d u$

Step 5

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

$\int \frac{\sin (u)}{3} d u$

Step 6

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

$\frac{1}{3} \int \sin (u) d u$

Step 7

The integral of $\sin (u)$ with respect to $u$ is $- \cos (u)$ .

$\frac{1}{3} (- \cos (u) + C)$

Step 8

Simplify.

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Step 8.1

Simplify.

$\frac{1}{3} (- \cos (u)) + C$

Step 8.2

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

$- \frac{\cos (u)}{3} + C$

Step 9

Replace all occurrences of $u$ with $3 x + \frac{π}{6}$ .

$- \frac{\cos (3 x + \frac{π}{6})}{3} + C$

Step 10

Reorder terms.

$- \frac{1}{3} \cos (3 x + \frac{π}{6}) + C$

Step 11

The answer is the antiderivative of the function $f (x) = \sin (3 x + \frac{π}{6})$ .

$F (x) =$ $- \frac{1}{3} \cos (3 x + \frac{π}{6}) + C$