Calculus Examples

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

Step 1

Factor $4$ out of $1 + 4 x^{2}$ .

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Step 1.1

Factor $4$ out of $1$ .

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

Step 1.2

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

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

Step 1.3

Factor $4$ out of $4 (\frac{1}{4}) + 4 (x^{2})$ .

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

Step 2

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

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

Step 3

Rewrite $\frac{1}{4}$ as ${(\frac{1}{2})}^{2}$ .

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

Step 4

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

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

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 $\frac{1}{2}$ .

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

Step 5.1.2

Multiply $2$ by $1$ .

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

Step 5.1.3

Multiply by the reciprocal of the fraction to divide by $\frac{1}{2}$ .

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

Step 5.1.4

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

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

Step 5.2

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

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

Step 5.3

Simplify.

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Step 5.3.1

Combine $\frac{1}{4}$ and $2$ .

$\frac{2}{4} \arctan (2 x) + C$

Step 5.3.2

Cancel the common factor of $2$ and $4$ .

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Step 5.3.2.1

Factor $2$ out of $2$ .

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

Step 5.3.2.2

Cancel the common factors.

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Step 5.3.2.2.1

Factor $2$ out of $4$ .

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

Step 5.3.2.2.2

Cancel the common factor.

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

Step 5.3.2.2.3

Rewrite the expression.

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