Calculus Examples

$\int \frac{12}{1 + 9 x^{2}} d x$

Step 1

Since $12$ is constant with respect to $x$ , move $12$ out of the integral.

$12 \int \frac{1}{1 + 9 x^{2}} d x$

Step 2

Factor $9$ out of $1 + 9 x^{2}$ .

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Step 2.1

Factor $9$ out of $1$ .

$12 \int \frac{1}{9 (\frac{1}{9}) + 9 x^{2}} d x$

Step 2.2

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

$12 \int \frac{1}{9 (\frac{1}{9}) + 9 (x^{2})} d x$

Step 2.3

Factor $9$ out of $9 (\frac{1}{9}) + 9 (x^{2})$ .

$12 \int \frac{1}{9 (\frac{1}{9} + x^{2})} d x$

Step 3

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

$12 (\frac{1}{9} \int \frac{1}{\frac{1}{9} + x^{2}} d x)$

Step 4

Simplify.

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Step 4.1

Combine $\frac{1}{9}$ and $12$ .

$\frac{12}{9} \int \frac{1}{\frac{1}{9} + x^{2}} d x$

Step 4.2

Cancel the common factor of $12$ and $9$ .

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Step 4.2.1

Factor $3$ out of $12$ .

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

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

Factor $3$ out of $9$ .

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

Step 4.2.2.2

Cancel the common factor.

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

Step 4.2.2.3

Rewrite the expression.

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

Step 5

Rewrite $\frac{1}{9}$ as ${(\frac{1}{3})}^{2}$ .

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

Step 6

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

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

Step 7

Simplify the answer.

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Step 7.1

Simplify.

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Step 7.1.1

Multiply by the reciprocal of the fraction to divide by $\frac{1}{3}$ .

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

Step 7.1.2

Multiply $3$ by $1$ .

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

Step 7.1.3

Multiply by the reciprocal of the fraction to divide by $\frac{1}{3}$ .

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

Step 7.1.4

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

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

Step 7.2

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

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

Step 7.3

Simplify.

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Step 7.3.1

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

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

Step 7.3.2

Multiply $4$ by $3$ .

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

Step 7.3.3

Cancel the common factor of $12$ and $3$ .

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Step 7.3.3.1

Factor $3$ out of $12$ .

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

Step 7.3.3.2

Cancel the common factors.

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Step 7.3.3.2.1

Factor $3$ out of $3$ .

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

Step 7.3.3.2.2

Cancel the common factor.

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

Step 7.3.3.2.3

Rewrite the expression.

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

Step 7.3.3.2.4

Divide $4$ by $1$ .

$4 \arctan (3 x) + C$