Calculus Examples

$\int \frac{1}{9 x^{2} + 16} d x$

Step 1

Reorder $9 x^{2}$ and $16$ .

$\int \frac{1}{16 + 9 x^{2}} d x$

Step 2

Factor $9$ out of $16 + 9 x^{2}$ .

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Step 2.1

Factor $9$ out of $16$ .

$\int \frac{1}{9 (\frac{16}{9}) + 9 x^{2}} d x$

Step 2.2

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

$\int \frac{1}{9 (\frac{16}{9}) + 9 (x^{2})} d x$

Step 2.3

Factor $9$ out of $9 (\frac{16}{9}) + 9 (x^{2})$ .

$\int \frac{1}{9 (\frac{16}{9} + x^{2})} d x$

Step 3

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

$\frac{1}{9} \int \frac{1}{\frac{16}{9} + x^{2}} d x$

Step 4

Rewrite $\frac{16}{9}$ as ${(\frac{4}{3})}^{2}$ .

$\frac{1}{9} \int \frac{1}{{(\frac{4}{3})}^{2} + x^{2}} d x$

Step 5

The integral of $\frac{1}{{(\frac{4}{3})}^{2} + x^{2}}$ with respect to $x$ is $\frac{1}{\frac{4}{3}} \arctan (\frac{x}{\frac{4}{3}}) + C$ .

$\frac{1}{9} (\frac{1}{\frac{4}{3}} \arctan (\frac{x}{\frac{4}{3}}) + C)$

Step 6

Simplify the answer.

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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 $\frac{4}{3}$ .

$\frac{1}{9} (1 (\frac{3}{4}) \arctan (\frac{x}{\frac{4}{3}}) + C)$

Step 6.1.2

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

$\frac{1}{9} (\frac{3}{4} \arctan (\frac{x}{\frac{4}{3}}) + C)$

Step 6.1.3

Multiply by the reciprocal of the fraction to divide by $\frac{4}{3}$ .

$\frac{1}{9} (\frac{3}{4} \arctan (x \frac{3}{4}) + C)$

Step 6.1.4

Combine $x$ and $\frac{3}{4}$ .

$\frac{1}{9} (\frac{3}{4} \arctan (\frac{x \cdot 3}{4}) + C)$

Step 6.1.5

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

$\frac{1}{9} (\frac{3}{4} \arctan (\frac{3 \cdot x}{4}) + C)$

Step 6.1.6

Combine $\frac{3}{4}$ and $\arctan (\frac{3 x}{4})$ .

$\frac{1}{9} (\frac{3 \arctan (\frac{3 x}{4})}{4} + C)$

Step 6.2

Rewrite $\frac{1}{9} (\frac{3 \arctan (\frac{3 x}{4})}{4} + C)$ as $\frac{1}{9} \cdot \frac{3}{4} \arctan (\frac{3}{4} x) + C$ .

$\frac{1}{9} \cdot \frac{3}{4} \arctan (\frac{3}{4} x) + C$

Step 6.3

Simplify.

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Step 6.3.1

Multiply $\frac{1}{9}$ by $\frac{3}{4}$ .

$\frac{3}{9 \cdot 4} \arctan (\frac{3}{4} x) + C$

Step 6.3.2

Multiply $9$ by $4$ .

$\frac{3}{36} \arctan (\frac{3}{4} x) + C$

Step 6.3.3

Cancel the common factor of $3$ and $36$ .

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Step 6.3.3.1

Factor $3$ out of $3$ .

$\frac{3 (1)}{36} \arctan (\frac{3}{4} x) + C$

Step 6.3.3.2

Cancel the common factors.

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Step 6.3.3.2.1

Factor $3$ out of $36$ .

$\frac{3 \cdot 1}{3 \cdot 12} \arctan (\frac{3}{4} x) + C$

Step 6.3.3.2.2

Cancel the common factor.

$\frac{3 \cdot 1}{3 \cdot 12} \arctan (\frac{3}{4} x) + C$

Step 6.3.3.2.3

Rewrite the expression.

$\frac{1}{12} \arctan (\frac{3}{4} x) + C$