Calculus Examples

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

Step 1

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

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

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.

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

Step 4

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

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Step 4.1.1

Differentiate $1 + x^{2}$ .

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

Step 4.1.2

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

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

Step 4.1.3

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

$0 + \frac{d}{d x} [x^{2}]$

Step 4.1.4

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

$0 + 2 x$

Step 4.1.5

Add $0$ and $2 x$ .

$2 x$

Step 4.2

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

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

Step 5

Simplify.

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Step 5.1

Multiply $\frac{1}{\sqrt{u}}$ by $\frac{1}{2}$ .

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

Step 5.2

Move $2$ to the left of $\sqrt{u}$ .

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

Step 6

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

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

Step 7

Apply basic rules of exponents.

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Step 7.1

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

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

Step 7.2

Move $u^{\frac{1}{2}}$ out of the denominator by raising it to the $- 1$ power.

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

Step 7.3

Multiply the exponents in ${(u^{\frac{1}{2}})}^{- 1}$ .

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Step 7.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 7.3.2

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

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

Step 7.3.3

Move the negative in front of the fraction.

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

Step 8

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

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

Step 9

Simplify.

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Step 9.1

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

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

Step 9.2

Simplify.

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Step 9.2.1

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

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

Step 9.2.2

Cancel the common factor of $2$ .

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Step 9.2.2.1

Cancel the common factor.

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

Step 9.2.2.2

Rewrite the expression.

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

Step 9.2.3

Multiply $u^{\frac{1}{2}}$ by $1$ .

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

Step 10

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

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

Step 11

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

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