Calculus Examples

$\int \frac{x \sin (\sqrt{x^{2} + 4})}{\sqrt{x^{2} + 4}} d x$

Step 1

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

Tap for more steps...

Step 1.1

Let $u_{1} = x^{2} + 4$ . Find $\frac{d u_{1}}{d x}$ .

Tap for more steps...

Step 1.1.1

Differentiate $x^{2} + 4$ .

$\frac{d}{d x} [x^{2} + 4]$

Step 1.1.2

By the Sum Rule, the derivative of $x^{2} + 4$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [4]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [4]$

Step 1.1.3

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

$2 x + \frac{d}{d x} [4]$

Step 1.1.4

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$2 x + 0$

Step 1.1.5

Add $2 x$ and $0$ .

$2 x$

Step 1.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$\int \frac{\sin (\sqrt{u_{1}})}{\sqrt{u_{1}}} \cdot \frac{1}{2} d u_{1}$

Step 2

Simplify.

Tap for more steps...

Step 2.1

Multiply $\frac{\sin (\sqrt{u_{1}})}{\sqrt{u_{1}}}$ by $\frac{1}{2}$ .

$\int \frac{\sin (\sqrt{u_{1}})}{\sqrt{u_{1}} \cdot 2} d u_{1}$

Step 2.2

Move $2$ to the left of $\sqrt{u_{1}}$ .

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

Step 3

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

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

Step 4

Apply basic rules of exponents.

Tap for more steps...

Step 4.1

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

$\frac{1}{2} \int \frac{\sin ({u_{1}}^{\frac{1}{2}})}{\sqrt{u_{1}}} d u_{1}$

Step 4.2

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

$\frac{1}{2} \int \frac{\sin ({u_{1}}^{\frac{1}{2}})}{{u_{1}}^{\frac{1}{2}}} d u_{1}$

Step 4.3

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

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

Step 4.4

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

Tap for more steps...

Step 4.4.1

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

$\frac{1}{2} \int \sin ({u_{1}}^{\frac{1}{2}}) {u_{1}}^{\frac{1}{2} \cdot - 1} d u_{1}$

Step 4.4.2

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

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

Step 4.4.3

Move the negative in front of the fraction.

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

Step 5

Let $u_{2} = {u_{1}}^{\frac{1}{2}}$ . Then $d u_{2} = \frac{1}{2 {u_{1}}^{\frac{1}{2}}} d u_{1}$ , so $- d u_{2} = \frac{- 1}{2 {u_{1}}^{\frac{1}{2}}} d u_{1}$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

Tap for more steps...

Step 5.1

Let $u_{2} = {u_{1}}^{\frac{1}{2}}$ . Find $\frac{d u_{2}}{d u_{1}}$ .

Tap for more steps...

Step 5.1.1

Differentiate ${u_{1}}^{\frac{1}{2}}$ .

$\frac{d}{d u_{1}} [{u_{1}}^{\frac{1}{2}}]$

Step 5.1.2

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

$\frac{1}{2} {u_{1}}^{\frac{1}{2} - 1}$

Step 5.1.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{2} {u_{1}}^{\frac{1}{2} - 1 \cdot \frac{2}{2}}$

Step 5.1.4

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

$\frac{1}{2} {u_{1}}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}$

Step 5.1.5

Combine the numerators over the common denominator.

$\frac{1}{2} {u_{1}}^{\frac{1 - 1 \cdot 2}{2}}$

Step 5.1.6

Simplify the numerator.

Tap for more steps...

Step 5.1.6.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} {u_{1}}^{\frac{1 - 2}{2}}$

Step 5.1.6.2

Subtract $2$ from $1$ .

$\frac{1}{2} {u_{1}}^{\frac{- 1}{2}}$

Step 5.1.7

Move the negative in front of the fraction.

$\frac{1}{2} {u_{1}}^{- \frac{1}{2}}$

Step 5.1.8

Simplify.

Tap for more steps...

Step 5.1.8.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2} \cdot \frac{1}{{u_{1}}^{\frac{1}{2}}}$

Step 5.1.8.2

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

$\frac{1}{2 {u_{1}}^{\frac{1}{2}}}$

Step 5.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\frac{1}{2} \int 2 \sin (u_{2}) d u_{2}$

Step 6

Since $2$ is constant with respect to $u_{2}$ , move $2$ out of the integral.

$\frac{1}{2} (2 \int \sin (u_{2}) d u_{2})$

Step 7

Simplify.

Tap for more steps...

Step 7.1

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

$\frac{2}{2} \int \sin (u_{2}) d u_{2}$

Step 7.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 7.2.1

Cancel the common factor.

$\frac{2}{2} \int \sin (u_{2}) d u_{2}$

Step 7.2.2

Rewrite the expression.

$1 \int \sin (u_{2}) d u_{2}$

Step 7.3

Multiply $\int \sin (u_{2}) d u_{2}$ by $1$ .

$\int \sin (u_{2}) d u_{2}$

Step 8

The integral of $\sin (u_{2})$ with respect to $u_{2}$ is $- \cos (u_{2})$ .

$- \cos (u_{2}) + C$

Step 9

Substitute back in for each integration substitution variable.

Tap for more steps...

Step 9.1

Replace all occurrences of $u_{2}$ with ${u_{1}}^{\frac{1}{2}}$ .

$- \cos ({u_{1}}^{\frac{1}{2}}) + C$

Step 9.2

Replace all occurrences of $u_{1}$ with $x^{2} + 4$ .

$- \cos ({(x^{2} + 4)}^{\frac{1}{2}}) + C$