Calculus Examples

$\int \frac{\sin (\sqrt{3 x})}{\sqrt{3 x}} d x$

Step 1

Let $u_{1} = 3 x$ . Then $d u_{1} = 3 d x$ , so $\frac{1}{3} d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 1.1

Let $u_{1} = 3 x$ . Find $\frac{d u_{1}}{d x}$ .

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Step 1.1.1

Differentiate $3 x$ .

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

Step 1.1.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

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

Step 1.1.3

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

$3 \cdot 1$

Step 1.1.4

Multiply $3$ by $1$ .

$3$

Step 1.2

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

$\int \frac{\sin (\sqrt{u_{1}})}{\sqrt{u_{1}}} \cdot \frac{1}{3} d u_{1}$

Step 2

Simplify.

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Step 2.1

Multiply $\frac{\sin (\sqrt{u_{1}})}{\sqrt{u_{1}}}$ by $\frac{1}{3}$ .

$\int \frac{\sin (\sqrt{u_{1}})}{\sqrt{u_{1}} \cdot 3} d u_{1}$

Step 2.2

Move $3$ to the left of $\sqrt{u_{1}}$ .

$\int \frac{\sin (\sqrt{u_{1}})}{3 \sqrt{u_{1}}} d u_{1}$

Step 3

Since $\frac{1}{3}$ is constant with respect to $u_{1}$ , move $\frac{1}{3}$ out of the integral.

$\frac{1}{3} \int \frac{\sin (\sqrt{u_{1}})}{\sqrt{u_{1}}} d u_{1}$

Step 4

Apply basic rules of exponents.

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Step 4.1

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

$\frac{1}{3} \int \frac{\sin ({u_{1}}^{\frac{1}{2}})}{\sqrt{u_{1}}} d u_{1}$

Step 4.2

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

$\frac{1}{3} \int \frac{\sin ({u_{1}}^{\frac{1}{2}})}{{u_{1}}^{\frac{1}{2}}} d u_{1}$

Step 4.3

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

$\frac{1}{3} \int \sin ({u_{1}}^{\frac{1}{2}}) {({u_{1}}^{\frac{1}{2}})}^{- 1} d u_{1}$

Step 4.4

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

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Step 4.4.1

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

$\frac{1}{3} \int \sin ({u_{1}}^{\frac{1}{2}}) {u_{1}}^{\frac{1}{2} \cdot - 1} d u_{1}$

Step 4.4.2

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

$\frac{1}{3} \int \sin ({u_{1}}^{\frac{1}{2}}) {u_{1}}^{\frac{- 1}{2}} d u_{1}$

Step 4.4.3

Move the negative in front of the fraction.

$\frac{1}{3} \int \sin ({u_{1}}^{\frac{1}{2}}) {u_{1}}^{- \frac{1}{2}} d u_{1}$

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 $d$ $u_{2}$ .

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Step 5.1

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

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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.

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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.

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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 $d u_{2}$ .

$\frac{1}{3} \int 2 \sin (u_{2}) d u_{2}$

Step 6

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

$\frac{1}{3} (2 \int \sin (u_{2}) d u_{2})$

Step 7

Combine $2$ and $\frac{1}{3}$ .

$\frac{2}{3} \int \sin (u_{2}) d u_{2}$

Step 8

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

$\frac{2}{3} (- \cos (u_{2}) + C)$

Step 9

Simplify.

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Step 9.1

Simplify.

$\frac{2}{3} (- \cos (u_{2})) + C$

Step 9.2

Combine $\frac{2}{3}$ and $\cos (u_{2})$ .

$- \frac{2 \cos (u_{2})}{3} + C$

Step 10

Substitute back in for each integration substitution variable.

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Step 10.1

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

$- \frac{2 \cos ({u_{1}}^{\frac{1}{2}})}{3} + C$

Step 10.2

Replace all occurrences of $u_{1}$ with $3 x$ .

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

Step 11

Simplify.

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Step 11.1

Apply the product rule to $3 x$ .

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

Step 11.2

Reorder terms.

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