Calculus Examples

$\int x (2 x^{2} + 1) d x$

Step 1

Let $u = 2 x^{2} + 1$ . Then $d u = 4 x d x$ , so $\frac{1}{4} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 2 x^{2} + 1$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $2 x^{2} + 1$ .

$\frac{d}{d x} [2 x^{2} + 1]$

Step 1.1.2

By the Sum Rule, the derivative of $2 x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [1]$

Step 1.1.3

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

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Step 1.1.3.1

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

$2 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$

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 $n = 2$ .

$2 (2 x) + \frac{d}{d x} [1]$

Step 1.1.3.3

Multiply $2$ by $2$ .

$4 x + \frac{d}{d x} [1]$

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$4 x + 0$

Step 1.1.4.2

Add $4 x$ and $0$ .

$4 x$

Step 1.2

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

$\int u \frac{1}{4} d u$

Step 2

Combine $u$ and $\frac{1}{4}$ .

$\int \frac{u}{4} d u$

Step 3

Since $\frac{1}{4}$ is constant with respect to $u$ , move $\frac{1}{4}$ out of the integral.

$\frac{1}{4} \int u d u$

Step 4

By the Power Rule, the integral of $u$ with respect to $u$ is $\frac{1}{2} u^{2}$ .

$\frac{1}{4} (\frac{1}{2} u^{2} + C)$

Step 5

Simplify.

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Step 5.1

Rewrite $\frac{1}{4} (\frac{1}{2} u^{2} + C)$ as $\frac{1}{4} \cdot \frac{1}{2} u^{2} + C$ .

$\frac{1}{4} \cdot \frac{1}{2} u^{2} + C$

Step 5.2

Simplify.

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Step 5.2.1

Multiply $\frac{1}{4}$ by $\frac{1}{2}$ .

$\frac{1}{4 \cdot 2} u^{2} + C$

Step 5.2.2

Multiply $4$ by $2$ .

$\frac{1}{8} u^{2} + C$

Step 6

Replace all occurrences of $u$ with $2 x^{2} + 1$ .

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