Calculus Examples

$- 6 \sin (3 x) d x$

Step 1

Since $- 6$ is constant with respect to $x$ , move $- 6$ out of the integral.

$- 6 \int \sin (3 x) d x$

Step 2

Let $u = 3 x$ . 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 2.1

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

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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 $u$ and $d u$ .

$- 6 \int \sin (u) \frac{1}{3} d u$

Step 3

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

$- 6 \int \frac{\sin (u)}{3} d u$

Step 4

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

$- 6 (\frac{1}{3} \int \sin (u) d u)$

Step 5

Simplify.

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Step 5.1

Combine $\frac{1}{3}$ and $- 6$ .

$\frac{- 6}{3} \int \sin (u) d u$

Step 5.2

Cancel the common factor of $- 6$ and $3$ .

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Step 5.2.1

Factor $3$ out of $- 6$ .

$\frac{3 \cdot - 2}{3} \int \sin (u) d u$

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot - 2}{3 (1)} \int \sin (u) d u$

Step 5.2.2.2

Cancel the common factor.

$\frac{3 \cdot - 2}{3 \cdot 1} \int \sin (u) d u$

Step 5.2.2.3

Rewrite the expression.

$\frac{- 2}{1} \int \sin (u) d u$

Step 5.2.2.4

Divide $- 2$ by $1$ .

$- 2 \int \sin (u) d u$

Step 6

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

$- 2 (- \cos (u) + C)$

Step 7

Simplify.

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Step 7.1

Simplify.

$- 2 (- \cos (u)) + C$

Step 7.2

Multiply $- 1$ by $- 2$ .

$2 \cos (u) + C$

Step 8

Replace all occurrences of $u$ with $3 x$ .

$2 \cos (3 x) + C$