Calculus Examples

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

Step 1

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

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Step 1.1

Factor $9$ out of $4$ .

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

Step 1.2

Factor $9$ out of $9 x^{2}$ .

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

Step 1.3

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

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

Step 2

Since $\frac{1}{9}$ is constant with respect to $x$ , move $\frac{1}{9}$ out of the integral.

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

Step 3

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

$\frac{1}{9} \int \frac{1}{{(\frac{2}{3})}^{2} + x^{2}} d x$

Step 4

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

$\frac{1}{9} (\frac{1}{\frac{2}{3}} \arctan (\frac{x}{\frac{2}{3}}) + C)$

Step 5

Simplify the answer.

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Step 5.1

Simplify.

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Step 5.1.1

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

$\frac{1}{9} (1 (\frac{3}{2}) \arctan (\frac{x}{\frac{2}{3}}) + C)$

Step 5.1.2

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

$\frac{1}{9} (\frac{3}{2} \arctan (\frac{x}{\frac{2}{3}}) + C)$

Step 5.1.3

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

$\frac{1}{9} (\frac{3}{2} \arctan (x \frac{3}{2}) + C)$

Step 5.1.4

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

$\frac{1}{9} (\frac{3}{2} \arctan (\frac{x \cdot 3}{2}) + C)$

Step 5.1.5

Move $3$ to the left of $x$ .

$\frac{1}{9} (\frac{3}{2} \arctan (\frac{3 \cdot x}{2}) + C)$

Step 5.1.6

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

$\frac{1}{9} (\frac{3 \arctan (\frac{3 x}{2})}{2} + C)$

Step 5.2

Rewrite $\frac{1}{9} (\frac{3 \arctan (\frac{3 x}{2})}{2} + C)$ as $\frac{1}{9} \cdot \frac{3}{2} \arctan (\frac{3}{2} x) + C$ .

$\frac{1}{9} \cdot \frac{3}{2} \arctan (\frac{3}{2} x) + C$

Step 5.3

Simplify.

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Step 5.3.1

Multiply $\frac{1}{9}$ by $\frac{3}{2}$ .

$\frac{3}{9 \cdot 2} \arctan (\frac{3}{2} x) + C$

Step 5.3.2

Multiply $9$ by $2$ .

$\frac{3}{18} \arctan (\frac{3}{2} x) + C$

Step 5.3.3

Cancel the common factor of $3$ and $18$ .

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Step 5.3.3.1

Factor $3$ out of $3$ .

$\frac{3 (1)}{18} \arctan (\frac{3}{2} x) + C$

Step 5.3.3.2

Cancel the common factors.

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Step 5.3.3.2.1

Factor $3$ out of $18$ .

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

Step 5.3.3.2.2

Cancel the common factor.

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

Step 5.3.3.2.3

Rewrite the expression.

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