Calculus Examples

$\sin (\frac{π}{2} - 6 x)$

Step 1

Write $\sin (\frac{π}{2} - 6 x)$ as a function.

$f (x) = \sin (\frac{π}{2} - 6 x)$

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.

$F (x) = \int \sin (\frac{π}{2} - 6 x) d x$

Step 4

Let $u = \frac{π}{2} - 6 x$ . Then $d u = - 6 d x$ , so $- \frac{1}{6} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.1

Let $u = \frac{π}{2} - 6 x$ . Find $\frac{d u}{d x}$ .

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Step 4.1.1

Differentiate $\frac{π}{2} - 6 x$ .

$\frac{d}{d x} [\frac{π}{2} - 6 x]$

Step 4.1.2

Differentiate.

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Step 4.1.2.1

By the Sum Rule, the derivative of $\frac{π}{2} - 6 x$ with respect to $x$ is $\frac{d}{d x} [\frac{π}{2}] + \frac{d}{d x} [- 6 x]$ .

$\frac{d}{d x} [\frac{π}{2}] + \frac{d}{d x} [- 6 x]$

Step 4.1.2.2

Since $\frac{π}{2}$ is constant with respect to $x$ , the derivative of $\frac{π}{2}$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- 6 x]$

Step 4.1.3

Evaluate $\frac{d}{d x} [- 6 x]$ .

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Step 4.1.3.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x$ with respect to $x$ is $- 6 \frac{d}{d x} [x]$ .

$0 - 6 \frac{d}{d x} [x]$

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 $n = 1$ .

$0 - 6 \cdot 1$

Step 4.1.3.3

Multiply $- 6$ by $1$ .

$0 - 6$

Step 4.1.4

Subtract $6$ from $0$ .

$- 6$

Step 4.2

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

$\int \sin (u) \frac{1}{- 6} d u$

Step 5

Simplify.

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Step 5.1

Move the negative in front of the fraction.

$\int \sin (u) (- \frac{1}{6}) d u$

Step 5.2

Combine $\sin (u)$ and $\frac{1}{6}$ .

$\int - \frac{\sin (u)}{6} d u$

Step 6

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$- \int \frac{\sin (u)}{6} d u$

Step 7

Since $\frac{1}{6}$ is constant with respect to $u$ , move $\frac{1}{6}$ out of the integral.

$- (\frac{1}{6} \int \sin (u) d u)$

Step 8

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

$- \frac{1}{6} (- \cos (u) + C)$

Step 9

Simplify.

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Step 9.1

Simplify.

$- \frac{1}{6} (- \cos (u)) + C$

Step 9.2

Simplify.

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Step 9.2.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{1}{6}) \cos (u) + C$

Step 9.2.2

Multiply $\frac{1}{6}$ by $1$ .

$\frac{1}{6} \cos (u) + C$

Step 9.2.3

Combine $\frac{1}{6}$ and $\cos (u)$ .

$\frac{\cos (u)}{6} + C$

Step 10

Replace all occurrences of $u$ with $\frac{π}{2} - 6 x$ .

$\frac{\cos (\frac{π}{2} - 6 x)}{6} + C$

Step 11

Reorder terms.

$\frac{1}{6} \cos (\frac{π}{2} - 6 x) + C$

Step 12

The answer is the antiderivative of the function $f (x) = \sin (\frac{π}{2} - 6 x)$ .

$F (x) =$ $\frac{1}{6} \cos (\frac{π}{2} - 6 x) + C$