Calculus Examples

$\int (2 x + 5) {(x^{2} + 5 x)}^{7} d x$

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $x^{2} + 5 x$ .

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

Step 1.1.2

Differentiate.

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Step 1.1.2.1

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

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

Step 1.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

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

Step 1.1.3

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

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Step 1.1.3.1

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

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

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 $n = 1$ .

$2 x + 5 \cdot 1$

Step 1.1.3.3

Multiply $5$ by $1$ .

$2 x + 5$

Step 1.2

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

$\int u^{7} d u$

Step 2

By the Power Rule, the integral of $u^{7}$ with respect to $u$ is $\frac{1}{8} u^{8}$ .

$\frac{1}{8} u^{8} + C$

Step 3

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

$\frac{1}{8} {(x^{2} + 5 x)}^{8} + C$