Calculus Examples

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

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

Step 1

Write

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

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

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

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

Step 2

The function

F (x)

$F (x)$ can be found by finding the indefinite integral of the derivative

f (x)

$f (x)$ .

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

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

Step 4

Let

u = 1 + x^{2}

$u = 1 + x^{2}$ . Then

d u = 2 x d x

$d u = 2 x d x$ , so

\frac{1}{2} d u = x d x

$\frac{1}{2} d u = x d x$ . Rewrite using

u

$u$ and

d

$d$

u

$u$ .

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Step 4.1

Let

u = 1 + x^{2}

$u = 1 + x^{2}$ . Find

\frac{d u}{d x}

$\frac{d u}{d x}$ .

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Step 4.1.1

Differentiate

1 + x^{2}

$1 + x^{2}$ .

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

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

Step 4.1.2

By the Sum Rule, the derivative of

1 + x^{2}

$1 + x^{2}$ with respect to

x

$x$ is

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

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

\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

$1$ is constant with respect to

x

$x$ , the derivative of

1

$1$ with respect to

x

$x$ is

0

$0$ .

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

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

Step 4.1.4

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 2

$n = 2$ .

0 + 2 x

$0 + 2 x$

Step 4.1.5

Add

0

$0$ and

2 x

$2 x$ .

2 x

$2 x$

2 x

$2 x$

Step 4.2

Rewrite the problem using

u

$u$ and

d u

$d u$ .

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

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

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

$\frac{1}{\sqrt{u}}$ by

\frac{1}{2}

$\frac{1}{2}$ .

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

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

Step 5.2

Move

2

$2$ to the left of

\sqrt{u}

$\sqrt{u}$ .

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

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

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

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

Step 6

Since

\frac{1}{2}

$\frac{1}{2}$ is constant with respect to

u

$u$ , move

\frac{1}{2}

$\frac{1}{2}$ out of the integral.

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

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

Step 7

Apply basic rules of exponents.

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Step 7.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{u}

$\sqrt{u}$ as

u^{\frac{1}{2}}

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

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

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

Step 7.2

Move

u^{\frac{1}{2}}

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

- 1

$- 1$ power.

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

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

Step 7.3

Multiply the exponents in

{(u^{\frac{1}{2}})}^{- 1}

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

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Step 7.3.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

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

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

Step 7.3.2

Combine

\frac{1}{2}

$\frac{1}{2}$ and

- 1

$- 1$ .

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

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

Step 7.3.3

Move the negative in front of the fraction.

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

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

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

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

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

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

Step 8

By the Power Rule, the integral of

u^{- \frac{1}{2}}

$u^{- \frac{1}{2}}$ with respect to

u

$u$ is

2 u^{\frac{1}{2}}

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

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

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

Step 9

Simplify.

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Step 9.1

Rewrite

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

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

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

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

\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}

$\frac{1}{2}$ and

2

$2$ .

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

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

Step 9.2.2

Cancel the common factor of

2

$2$ .

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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

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

$F (x) =$

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