Calculus Examples

$\int 2 x \sqrt{x^{2} + 1} d x$

Step 1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int x \sqrt{x^{2} + 1} d x$

Step 2

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

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Step 2.1

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

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Step 2.1.1

Differentiate $x^{2} + 1$ .

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.1.4

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

$2 x + 0$

Step 2.1.5

Add $2 x$ and $0$ .

$2 x$

Step 2.2

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

$2 \int \sqrt{u} \frac{1}{2} d u$

Step 3

Combine $\sqrt{u}$ and $\frac{1}{2}$ .

$2 \int \frac{\sqrt{u}}{2} d u$

Step 4

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$2 (\frac{1}{2} \int \sqrt{u} d u)$

Step 5

Simplify the expression.

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Step 5.1

Simplify.

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Step 5.1.1

Combine $\frac{1}{2}$ and $2$ .

$\frac{2}{2} \int \sqrt{u} d u$

Step 5.1.2

Cancel the common factor of $2$ .

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Step 5.1.2.1

Cancel the common factor.

$\frac{2}{2} \int \sqrt{u} d u$

Step 5.1.2.2

Rewrite the expression.

$1 \int \sqrt{u} d u$

Step 5.1.3

Multiply $\int \sqrt{u} d u$ by $1$ .

$\int \sqrt{u} d u$

Step 5.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u}$ as $u^{\frac{1}{2}}$ .

$\int u^{\frac{1}{2}} d u$

Step 6

By the Power Rule, the integral of $u^{\frac{1}{2}}$ with respect to $u$ is $\frac{2}{3} u^{\frac{3}{2}}$ .

$\frac{2}{3} u^{\frac{3}{2}} + C$

Step 7

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

$\frac{2}{3} {(x^{2} + 1)}^{\frac{3}{2}} + C$