Calculus Examples

$\int \frac{\sec (x) \tan (x)}{9 + 4 \sec^{2} (x)} d x$

Step 1

Let $u = \sec (x)$ . Then $d u = \sec (x) \tan (x) d x$ , so $\frac{1}{\sec (x) \tan (x)} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = \sec (x)$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $\sec (x)$ .

$\frac{d}{d x} [\sec (x)]$

Step 1.1.2

The derivative of $\sec (x)$ with respect to $x$ is $\sec (x) \tan (x)$ .

$\sec (x) \tan (x)$

Step 1.2

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

$\int \frac{1}{9 + 4 u^{2}} d u$

Step 2

Factor $4$ out of $9 + 4 u^{2}$ .

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Step 2.1

Factor $4$ out of $9$ .

$\int \frac{1}{4 (\frac{9}{4}) + 4 u^{2}} d u$

Step 2.2

Factor $4$ out of $4 u^{2}$ .

$\int \frac{1}{4 (\frac{9}{4}) + 4 (u^{2})} d u$

Step 2.3

Factor $4$ out of $4 (\frac{9}{4}) + 4 (u^{2})$ .

$\int \frac{1}{4 (\frac{9}{4} + u^{2})} d u$

Step 3

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

$\frac{1}{4} \int \frac{1}{\frac{9}{4} + u^{2}} d u$

Step 4

Rewrite $\frac{9}{4}$ as ${(\frac{3}{2})}^{2}$ .

$\frac{1}{4} \int \frac{1}{{(\frac{3}{2})}^{2} + u^{2}} d u$

Step 5

The integral of $\frac{1}{{(\frac{3}{2})}^{2} + u^{2}}$ with respect to $u$ is $\frac{1}{\frac{3}{2}} \arctan (\frac{u}{\frac{3}{2}}) + C$ .

$\frac{1}{4} (\frac{1}{\frac{3}{2}} \arctan (\frac{u}{\frac{3}{2}}) + C)$

Step 6

Simplify.

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Step 6.1

Simplify.

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Step 6.1.1

Multiply by the reciprocal of the fraction to divide by $\frac{3}{2}$ .

$\frac{1}{4} (1 (\frac{2}{3}) \arctan (\frac{u}{\frac{3}{2}}) + C)$

Step 6.1.2

Multiply $\frac{2}{3}$ by $1$ .

$\frac{1}{4} (\frac{2}{3} \arctan (\frac{u}{\frac{3}{2}}) + C)$

Step 6.1.3

Multiply by the reciprocal of the fraction to divide by $\frac{3}{2}$ .

$\frac{1}{4} (\frac{2}{3} \arctan (u \frac{2}{3}) + C)$

Step 6.1.4

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

$\frac{1}{4} (\frac{2}{3} \arctan (\frac{u \cdot 2}{3}) + C)$

Step 6.1.5

Move $2$ to the left of $u$ .

$\frac{1}{4} (\frac{2}{3} \arctan (\frac{2 \cdot u}{3}) + C)$

Step 6.1.6

Combine $\frac{2}{3}$ and $\arctan (\frac{2 u}{3})$ .

$\frac{1}{4} (\frac{2 \arctan (\frac{2 u}{3})}{3} + C)$

Step 6.2

Rewrite $\frac{1}{4} (\frac{2 \arctan (\frac{2 u}{3})}{3} + C)$ as $\frac{1}{4} \cdot \frac{2}{3} \arctan (\frac{2}{3} u) + C$ .

$\frac{1}{4} \cdot \frac{2}{3} \arctan (\frac{2}{3} u) + C$

Step 6.3

Simplify.

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Step 6.3.1

Multiply $\frac{1}{4}$ by $\frac{2}{3}$ .

$\frac{2}{4 \cdot 3} \arctan (\frac{2}{3} u) + C$

Step 6.3.2

Multiply $4$ by $3$ .

$\frac{2}{12} \arctan (\frac{2}{3} u) + C$

Step 6.3.3

Cancel the common factor of $2$ and $12$ .

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Step 6.3.3.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{12} \arctan (\frac{2}{3} u) + C$

Step 6.3.3.2

Cancel the common factors.

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Step 6.3.3.2.1

Factor $2$ out of $12$ .

$\frac{2 \cdot 1}{2 \cdot 6} \arctan (\frac{2}{3} u) + C$

Step 6.3.3.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 6} \arctan (\frac{2}{3} u) + C$

Step 6.3.3.2.3

Rewrite the expression.

$\frac{1}{6} \arctan (\frac{2}{3} u) + C$

Step 7

Replace all occurrences of $u$ with $\sec (x)$ .

$\frac{1}{6} \arctan (\frac{2}{3} \sec (x)) + C$