Calculus Examples

$\frac{5}{9 x^{2} + 36}$

Step 1

Write $\frac{5}{9 x^{2} + 36}$ as a function.

$f (x) = \frac{5}{9 x^{2} + 36}$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int \frac{5}{9 x^{2} + 36} d x$

Step 4

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$5 \int \frac{1}{9 x^{2} + 36} d x$

Step 5

Reorder $9 x^{2}$ and $36$ .

$5 \int \frac{1}{36 + 9 x^{2}} d x$

Step 6

Factor $9$ out of $36 + 9 x^{2}$ .

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Step 6.1

Factor $9$ out of $36$ .

$5 \int \frac{1}{9 \cdot 4 + 9 x^{2}} d x$

Step 6.2

Factor $9$ out of $9 \cdot 4$ .

$5 \int \frac{1}{9 (4) + 9 x^{2}} d x$

Step 6.3

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

$5 \int \frac{1}{9 (4) + 9 (x^{2})} d x$

Step 6.4

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

$5 \int \frac{1}{9 (4 + x^{2})} d x$

Step 7

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

$5 (\frac{1}{9} \int \frac{1}{4 + x^{2}} d x)$

Step 8

Combine fractions.

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Step 8.1

Combine $\frac{1}{9}$ and $5$ .

$\frac{5}{9} \int \frac{1}{4 + x^{2}} d x$

Step 8.2

Rewrite $4$ as $2^{2}$ .

$\frac{5}{9} \int \frac{1}{2^{2} + x^{2}} d x$

Step 9

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

$\frac{5}{9} (\frac{1}{2} \arctan (\frac{x}{2}) + C)$

Step 10

Simplify the answer.

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Step 10.1

Combine $\frac{1}{2}$ and $\arctan (\frac{x}{2})$ .

$\frac{5}{9} (\frac{\arctan (\frac{x}{2})}{2} + C)$

Step 10.2

Rewrite $\frac{5}{9} (\frac{\arctan (\frac{x}{2})}{2} + C)$ as $\frac{5}{9} \cdot \frac{1}{2} \arctan (\frac{1}{2} x) + C$ .

$\frac{5}{9} \cdot \frac{1}{2} \arctan (\frac{1}{2} x) + C$

Step 10.3

Simplify.

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Step 10.3.1

Multiply $\frac{5}{9}$ by $\frac{1}{2}$ .

$\frac{5}{9 \cdot 2} \arctan (\frac{1}{2} x) + C$

Step 10.3.2

Multiply $9$ by $2$ .

$\frac{5}{18} \arctan (\frac{1}{2} x) + C$

Step 11

The answer is the antiderivative of the function $f (x) = \frac{5}{9 x^{2} + 36}$ .

$F (x) =$ $\frac{5}{18} \arctan (\frac{1}{2} x) + C$