Calculus Examples

$\int \frac{1}{\cos^{2} (t) \sqrt[9]{8 + \tan (t)}} d t$

Step 1

Simplify.

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Step 1.1

Multiply by $1$ .

$\int \frac{1}{\cos^{2} (t) \cdot 1 \sqrt[9]{8 + \tan (t)}} d t$

Step 1.2

Separate fractions.

$\int \frac{1}{1 \sqrt[9]{8 + \tan (t)}} \cdot \frac{1}{\cos^{2} (t)} d t$

Step 1.3

Convert from $\frac{1}{\cos^{2} (t)}$ to $\sec^{2} (t)$ .

$\int \frac{1}{1 \sqrt[9]{8 + \tan (t)}} \sec^{2} (t) d t$

Step 1.4

Multiply $\sqrt[9]{8 + \tan (t)}$ by $1$ .

$\int \frac{1}{\sqrt[9]{8 + \tan (t)}} \sec^{2} (t) d t$

Step 1.5

Combine $\frac{1}{\sqrt[9]{8 + \tan (t)}}$ and $\sec^{2} (t)$ .

$\int \frac{\sec^{2} (t)}{\sqrt[9]{8 + \tan (t)}} d t$

Step 2

Let $u = 8 + \tan (t)$ . Then $d u = \sec^{2} (t) d t$ , so $\frac{1}{\sec^{2} (t)} d u = d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = 8 + \tan (t)$ . Find $\frac{d u}{d t}$ .

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Step 2.1.1

Differentiate $8 + \tan (t)$ .

$\frac{d}{d t} [8 + \tan (t)]$

Step 2.1.2

Differentiate.

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Step 2.1.2.1

By the Sum Rule, the derivative of $8 + \tan (t)$ with respect to $t$ is $\frac{d}{d t} [8] + \frac{d}{d t} [\tan (t)]$ .

$\frac{d}{d t} [8] + \frac{d}{d t} [\tan (t)]$

Step 2.1.2.2

Since $8$ is constant with respect to $t$ , the derivative of $8$ with respect to $t$ is $0$ .

$0 + \frac{d}{d t} [\tan (t)]$

Step 2.1.3

The derivative of $\tan (t)$ with respect to $t$ is $\sec^{2} (t)$ .

$0 + \sec^{2} (t)$

Step 2.1.4

Add $0$ and $\sec^{2} (t)$ .

$\sec^{2} (t)$

Step 2.2

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

$\int \frac{1}{\sqrt[9]{u}} d u$

Step 3

Apply basic rules of exponents.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[9]{u}$ as $u^{\frac{1}{9}}$ .

$\int \frac{1}{u^{\frac{1}{9}}} d u$

Step 3.2

Move $u^{\frac{1}{9}}$ out of the denominator by raising it to the $- 1$ power.

$\int {(u^{\frac{1}{9}})}^{- 1} d u$

Step 3.3

Multiply the exponents in ${(u^{\frac{1}{9}})}^{- 1}$ .

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Step 3.3.1

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

$\int u^{\frac{1}{9} \cdot - 1} d u$

Step 3.3.2

Combine $\frac{1}{9}$ and $- 1$ .

$\int u^{\frac{- 1}{9}} d u$

Step 3.3.3

Move the negative in front of the fraction.

$\int u^{- \frac{1}{9}} d u$

Step 4

By the Power Rule, the integral of $u^{- \frac{1}{9}}$ with respect to $u$ is $\frac{9}{8} u^{\frac{8}{9}}$ .

$\frac{9}{8} u^{\frac{8}{9}} + C$

Step 5

Replace all occurrences of $u$ with $8 + \tan (t)$ .

$\frac{9}{8} {(8 + \tan (t))}^{\frac{8}{9}} + C$