Calculus Examples

$x \cdot \sqrt{x^{2} + 1}$

Step 1

Write $x \cdot \sqrt{x^{2} + 1}$ as a function.

$f (x) = x \cdot \sqrt{x^{2} + 1}$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int x \cdot \sqrt{x^{2} + 1} d x$

Step 4

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 4.1

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

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Step 4.1.1

Differentiate $x^{2} + 1$ .

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

Step 4.1.2

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 4.1.3

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 4.1.4

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

$2 x + 0$

Step 4.1.5

Add $2 x$ and $0$ .

$2 x$

Step 4.2

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Simplify.

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Step 9.1

Rewrite $\frac{1}{2} (\frac{2}{3} u^{\frac{3}{2}} + C)$ as $\frac{1}{2} \cdot \frac{2}{3} u^{\frac{3}{2}} + C$ .

$\frac{1}{2} \cdot \frac{2}{3} u^{\frac{3}{2}} + C$

Step 9.2

Simplify.

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Step 9.2.1

Multiply $\frac{1}{2}$ by $\frac{2}{3}$ .

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

Step 9.2.2

Multiply $2$ by $3$ .

$\frac{2}{6} u^{\frac{3}{2}} + C$

Step 9.2.3

Cancel the common factor of $2$ and $6$ .

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Step 9.2.3.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{6} u^{\frac{3}{2}} + C$

Step 9.2.3.2

Cancel the common factors.

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Step 9.2.3.2.1

Factor $2$ out of $6$ .

$\frac{2 \cdot 1}{2 \cdot 3} u^{\frac{3}{2}} + C$

Step 9.2.3.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 3} u^{\frac{3}{2}} + C$

Step 9.2.3.2.3

Rewrite the expression.

$\frac{1}{3} u^{\frac{3}{2}} + C$

Step 10

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

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

Step 11

The answer is the antiderivative of the function $f (x) = x \cdot \sqrt{x^{2} + 1}$ .

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