Calculus Examples

$\int \sin (- 5 x^{4} - x - 6) (- 20 x^{3} - 1) d x$

Step 1

Let $u = - 5 x^{4} - x - 6$ . Then $d u = (- 20 x^{3} - 1) d x$ , so $\frac{1}{- 20 x^{3} - 1} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = - 5 x^{4} - x - 6$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $- 5 x^{4} - x - 6$ .

$\frac{d}{d x} [- 5 x^{4} - x - 6]$

Step 1.1.2

By the Sum Rule, the derivative of $- 5 x^{4} - x - 6$ with respect to $x$ is $\frac{d}{d x} [- 5 x^{4}] + \frac{d}{d x} [- x] + \frac{d}{d x} [- 6]$ .

$\frac{d}{d x} [- 5 x^{4}] + \frac{d}{d x} [- x] + \frac{d}{d x} [- 6]$

Step 1.1.3

Evaluate $\frac{d}{d x} [- 5 x^{4}]$ .

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Step 1.1.3.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x^{4}$ with respect to $x$ is $- 5 \frac{d}{d x} [x^{4}]$ .

$- 5 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- x] + \frac{d}{d x} [- 6]$

Step 1.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$- 5 (4 x^{3}) + \frac{d}{d x} [- x] + \frac{d}{d x} [- 6]$

Step 1.1.3.3

Multiply $4$ by $- 5$ .

$- 20 x^{3} + \frac{d}{d x} [- x] + \frac{d}{d x} [- 6]$

Step 1.1.4

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

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Step 1.1.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$- 20 x^{3} - \frac{d}{d x} [x] + \frac{d}{d x} [- 6]$

Step 1.1.4.2

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

$- 20 x^{3} - 1 \cdot 1 + \frac{d}{d x} [- 6]$

Step 1.1.4.3

Multiply $- 1$ by $1$ .

$- 20 x^{3} - 1 + \frac{d}{d x} [- 6]$

Step 1.1.5

Differentiate using the Constant Rule.

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Step 1.1.5.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$- 20 x^{3} - 1 + 0$

Step 1.1.5.2

Add $- 20 x^{3} - 1$ and $0$ .

$- 20 x^{3} - 1$

Step 1.2

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

$\int \sin (u) d u$

Step 2

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

$- \cos (u) + C$

Step 3

Replace all occurrences of $u$ with $- 5 x^{4} - x - 6$ .

$- \cos (- 5 x^{4} - x - 6) + C$