Calculus Examples

$\int x^{2} {(x^{3} + 5)}^{9} d x$

Step 1

Let $u = x^{3} + 5$ . Then $d u = 3 x^{2} d x$ , so $\frac{1}{3} d u = x^{2} d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = x^{3} + 5$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $x^{3} + 5$ .

$\frac{d}{d x} [x^{3} + 5]$

Step 1.1.2

By the Sum Rule, the derivative of $x^{3} + 5$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [5]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [5]$

Step 1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$3 x^{2} + \frac{d}{d x} [5]$

Step 1.1.4

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$3 x^{2} + 0$

Step 1.1.5

Add $3 x^{2}$ and $0$ .

$3 x^{2}$

Step 1.2

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

$\int u^{9} \frac{1}{3} d u$

Step 2

Combine $u^{9}$ and $\frac{1}{3}$ .

$\int \frac{u^{9}}{3} d u$

Step 3

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

$\frac{1}{3} \int u^{9} d u$

Step 4

By the Power Rule, the integral of $u^{9}$ with respect to $u$ is $\frac{1}{10} u^{10}$ .

$\frac{1}{3} (\frac{1}{10} u^{10} + C)$

Step 5

Simplify.

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Step 5.1

Rewrite $\frac{1}{3} (\frac{1}{10} u^{10} + C)$ as $\frac{1}{3} \cdot \frac{1}{10} u^{10} + C$ .

$\frac{1}{3} \cdot \frac{1}{10} u^{10} + C$

Step 5.2

Simplify.

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Step 5.2.1

Multiply $\frac{1}{3}$ by $\frac{1}{10}$ .

$\frac{1}{3 \cdot 10} u^{10} + C$

Step 5.2.2

Multiply $3$ by $10$ .

$\frac{1}{30} u^{10} + C$

Step 6

Replace all occurrences of $u$ with $x^{3} + 5$ .

$\frac{1}{30} {(x^{3} + 5)}^{10} + C$