Calculus Examples

$\int \frac{8 \sin^{3} (\sqrt{x})}{\sqrt{x}} d x$

Step 1

Since $8$ is constant with respect to $x$ , move $8$ out of the integral.

$8 \int \frac{\sin^{3} (\sqrt{x})}{\sqrt{x}} d x$

Step 2

Apply basic rules of exponents.

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Step 2.1

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

$8 \int \frac{\sin^{3} (x^{\frac{1}{2}})}{\sqrt{x}} d x$

Step 2.2

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

$8 \int \frac{\sin^{3} (x^{\frac{1}{2}})}{x^{\frac{1}{2}}} d x$

Step 2.3

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

$8 \int \sin^{3} (x^{\frac{1}{2}}) {(x^{\frac{1}{2}})}^{- 1} d x$

Step 2.4

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

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Step 2.4.1

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

$8 \int \sin^{3} (x^{\frac{1}{2}}) x^{\frac{1}{2} \cdot - 1} d x$

Step 2.4.2

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

$8 \int \sin^{3} (x^{\frac{1}{2}}) x^{\frac{- 1}{2}} d x$

Step 2.4.3

Move the negative in front of the fraction.

$8 \int \sin^{3} (x^{\frac{1}{2}}) x^{- \frac{1}{2}} d x$

Step 3

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

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.1.4

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

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

Step 3.1.5

Combine the numerators over the common denominator.

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

Step 3.1.6

Simplify the numerator.

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Step 3.1.6.1

Multiply $- 1$ by $2$ .

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

Step 3.1.6.2

Subtract $2$ from $1$ .

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

Step 3.1.7

Move the negative in front of the fraction.

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

Step 3.1.8

Simplify.

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Step 3.1.8.1

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

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

Step 3.1.8.2

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

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

Step 3.2

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

$8 \int 2 \sin^{3} (u_{1}) d u_{1}$

Step 4

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

$8 (2 \int \sin^{3} (u_{1}) d u_{1})$

Step 5

Multiply $2$ by $8$ .

$16 \int \sin^{3} (u_{1}) d u_{1}$

Step 6

Factor out $\sin^{2} (u_{1})$ .

$16 \int \sin^{2} (u_{1}) \sin (u_{1}) d u_{1}$

Step 7

Using the Pythagorean Identity, rewrite $\sin^{2} (u_{1})$ as $1 - \cos^{2} (u_{1})$ .

$16 \int (1 - \cos^{2} (u_{1})) \sin (u_{1}) d u_{1}$

Step 8

Let $u_{2} = \cos (u_{1})$ . Then $d u_{2} = - \sin (u_{1}) d u_{1}$ , so $- \frac{1}{\sin (u_{1})} d u_{2} = d u_{1}$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 8.1

Let $u_{2} = \cos (u_{1})$ . Find $\frac{d u_{2}}{d u_{1}}$ .

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Step 8.1.1

Differentiate $\cos (u_{1})$ .

$\frac{d}{d u_{1}} [\cos (u_{1})]$

Step 8.1.2

The derivative of $\cos (u_{1})$ with respect to $u_{1}$ is $- \sin (u_{1})$ .

$- \sin (u_{1})$

Step 8.2

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

$16 \int - 1 + {u_{2}}^{2} d u_{2}$

Step 9

Split the single integral into multiple integrals.

$16 (\int - 1 d u_{2} + \int {u_{2}}^{2} d u_{2})$

Step 10

Apply the constant rule.

$16 (- u_{2} + C + \int {u_{2}}^{2} d u_{2})$

Step 11

By the Power Rule, the integral of ${u_{2}}^{2}$ with respect to $u_{2}$ is $\frac{1}{3} {u_{2}}^{3}$ .

$16 (- u_{2} + C + \frac{1}{3} {u_{2}}^{3} + C)$

Step 12

Simplify.

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Step 12.1

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

$16 (- u_{2} + C + \frac{{u_{2}}^{3}}{3} + C)$

Step 12.2

Simplify.

$16 (- u_{2} + \frac{1}{3} {u_{2}}^{3}) + C$

Step 13

Substitute back in for each integration substitution variable.

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Step 13.1

Replace all occurrences of $u_{2}$ with $\cos (u_{1})$ .

$16 (- \cos (u_{1}) + \frac{1}{3} \cos^{3} (u_{1})) + C$

Step 13.2

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

$16 (- \cos (x^{\frac{1}{2}}) + \frac{1}{3} \cos^{3} (x^{\frac{1}{2}})) + C$

Step 14

Simplify.

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Step 14.1

Combine $\frac{1}{3}$ and $\cos^{3} (x^{\frac{1}{2}})$ .

$16 (- \cos (x^{\frac{1}{2}}) + \frac{\cos^{3} (x^{\frac{1}{2}})}{3}) + C$

Step 14.2

Apply the distributive property.

$16 (- \cos (x^{\frac{1}{2}})) + 16 \frac{\cos^{3} (x^{\frac{1}{2}})}{3} + C$

Step 14.3

Multiply $- 1$ by $16$ .

$- 16 \cos (x^{\frac{1}{2}}) + 16 \frac{\cos^{3} (x^{\frac{1}{2}})}{3} + C$

Step 14.4

Combine $16$ and $\frac{\cos^{3} (x^{\frac{1}{2}})}{3}$ .

$- 16 \cos (x^{\frac{1}{2}}) + \frac{16 \cos^{3} (x^{\frac{1}{2}})}{3} + C$

Step 15

Reorder terms.

$- 16 \cos (x^{\frac{1}{2}}) + \frac{16}{3} \cos^{3} (x^{\frac{1}{2}}) + C$