Calculus Examples

$\int (10 x^{4} - 4) \cos (2 x^{5} - 4 x) d x$

Step 1

Let $u = 2 x^{5} - 4 x$ . Then $d u = (10 x^{4} - 4) d x$ , so $\frac{1}{10 x^{4} - 4} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 2 x^{5} - 4 x$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $2 x^{5} - 4 x$ .

$\frac{d}{d x} [2 x^{5} - 4 x]$

Step 1.1.2

By the Sum Rule, the derivative of $2 x^{5} - 4 x$ with respect to $x$ is $\frac{d}{d x} [2 x^{5}] + \frac{d}{d x} [- 4 x]$ .

$\frac{d}{d x} [2 x^{5}] + \frac{d}{d x} [- 4 x]$

Step 1.1.3

Evaluate $\frac{d}{d x} [2 x^{5}]$ .

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Step 1.1.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{5}$ with respect to $x$ is $2 \frac{d}{d x} [x^{5}]$ .

$2 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 4 x]$

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 $n = 5$ .

$2 (5 x^{4}) + \frac{d}{d x} [- 4 x]$

Step 1.1.3.3

Multiply $5$ by $2$ .

$10 x^{4} + \frac{d}{d x} [- 4 x]$

Step 1.1.4

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

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Step 1.1.4.1

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

$10 x^{4} - 4 \frac{d}{d x} [x]$

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 $n = 1$ .

$10 x^{4} - 4 \cdot 1$

Step 1.1.4.3

Multiply $- 4$ by $1$ .

$10 x^{4} - 4$

Step 1.2

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

$\int \cos (u) d u$

Step 2

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

$\sin (u) + C$

Step 3

Replace all occurrences of $u$ with $2 x^{5} - 4 x$ .

$\sin (2 x^{5} - 4 x) + C$