Calculus Examples

$\frac{x}{\sqrt{2 x^{2} + 3}}$

Step 1

Write $\frac{x}{\sqrt{2 x^{2} + 3}}$ as a function.

$f (x) = \frac{x}{\sqrt{2 x^{2} + 3}}$

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.

$F (x) = \int \frac{x}{\sqrt{2 x^{2} + 3}} d x$

Step 4

Let $u = 2 x^{2} + 3$ . Then $d u = 4 x d x$ , so $\frac{1}{4} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 4.1

Let $u = 2 x^{2} + 3$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 4.1.1

Differentiate $2 x^{2} + 3$ .

$\frac{d}{d x} [2 x^{2} + 3]$

Step 4.1.2

By the Sum Rule, the derivative of $2 x^{2} + 3$ with respect to $x$ is $\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [3]$ .

$\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [3]$

Step 4.1.3

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

Tap for more steps...

Step 4.1.3.1

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

$2 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [3]$

Step 4.1.3.2

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

$2 (2 x) + \frac{d}{d x} [3]$

Step 4.1.3.3

Multiply $2$ by $2$ .

$4 x + \frac{d}{d x} [3]$

Step 4.1.4

Differentiate using the Constant Rule.

Tap for more steps...

Step 4.1.4.1

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$4 x + 0$

Step 4.1.4.2

Add $4 x$ and $0$ .

$4 x$

Step 4.2

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

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

Step 5

Simplify.

Tap for more steps...

Step 5.1

Multiply $\frac{1}{\sqrt{u}}$ by $\frac{1}{4}$ .

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

Step 5.2

Move $4$ to the left of $\sqrt{u}$ .

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

Step 6

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

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

Step 7

Apply basic rules of exponents.

Tap for more steps...

Step 7.1

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

$\frac{1}{4} \int \frac{1}{u^{\frac{1}{2}}} d u$

Step 7.2

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

$\frac{1}{4} \int {(u^{\frac{1}{2}})}^{- 1} d u$

Step 7.3

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

Tap for more steps...

Step 7.3.1

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

$\frac{1}{4} \int u^{\frac{1}{2} \cdot - 1} d u$

Step 7.3.2

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

$\frac{1}{4} \int u^{\frac{- 1}{2}} d u$

Step 7.3.3

Move the negative in front of the fraction.

$\frac{1}{4} \int u^{- \frac{1}{2}} d u$

Step 8

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

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

Step 9

Simplify.

Tap for more steps...

Step 9.1

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

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

Step 9.2

Simplify.

Tap for more steps...

Step 9.2.1

Combine $\frac{1}{4}$ and $2$ .

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

Step 9.2.2

Cancel the common factor of $2$ and $4$ .

Tap for more steps...

Step 9.2.2.1

Factor $2$ out of $2$ .

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

Step 9.2.2.2

Cancel the common factors.

Tap for more steps...

Step 9.2.2.2.1

Factor $2$ out of $4$ .

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

Step 9.2.2.2.2

Cancel the common factor.

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

Step 9.2.2.2.3

Rewrite the expression.

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

Step 10

Replace all occurrences of $u$ with $2 x^{2} + 3$ .

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

Step 11

The answer is the antiderivative of the function $f (x) = \frac{x}{\sqrt{2 x^{2} + 3}}$ .

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