Calculus Examples

$\frac{1}{1 + 25 x^{2}}$

Step 1

Write $\frac{1}{1 + 25 x^{2}}$ as a function.

$f (x) = \frac{1}{1 + 25 x^{2}}$

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.

$F (x) = \int \frac{1}{1 + 25 x^{2}} d x$

Step 4

Factor $25$ out of $1 + 25 x^{2}$ .

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Step 4.1

Factor $25$ out of $1$ .

$\int \frac{1}{25 (\frac{1}{25}) + 25 x^{2}} d x$

Step 4.2

Factor $25$ out of $25 x^{2}$ .

$\int \frac{1}{25 (\frac{1}{25}) + 25 (x^{2})} d x$

Step 4.3

Factor $25$ out of $25 (\frac{1}{25}) + 25 (x^{2})$ .

$\int \frac{1}{25 (\frac{1}{25} + x^{2})} d x$

Step 5

Since $\frac{1}{25}$ is constant with respect to $x$ , move $\frac{1}{25}$ out of the integral.

$\frac{1}{25} \int \frac{1}{\frac{1}{25} + x^{2}} d x$

Step 6

Rewrite $\frac{1}{25}$ as ${(\frac{1}{5})}^{2}$ .

$\frac{1}{25} \int \frac{1}{{(\frac{1}{5})}^{2} + x^{2}} d x$

Step 7

The integral of $\frac{1}{{(\frac{1}{5})}^{2} + x^{2}}$ with respect to $x$ is $\frac{1}{\frac{1}{5}} \arctan (\frac{x}{\frac{1}{5}}) + C$ .

$\frac{1}{25} (\frac{1}{\frac{1}{5}} \arctan (\frac{x}{\frac{1}{5}}) + C)$

Step 8

Simplify the answer.

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Step 8.1

Simplify.

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Step 8.1.1

Multiply by the reciprocal of the fraction to divide by $\frac{1}{5}$ .

$\frac{1}{25} (1 \cdot 5 \arctan (\frac{x}{\frac{1}{5}}) + C)$

Step 8.1.2

Multiply $5$ by $1$ .

$\frac{1}{25} (5 \arctan (\frac{x}{\frac{1}{5}}) + C)$

Step 8.1.3

Multiply by the reciprocal of the fraction to divide by $\frac{1}{5}$ .

$\frac{1}{25} (5 \arctan (x \cdot 5) + C)$

Step 8.1.4

Move $5$ to the left of $x$ .

$\frac{1}{25} (5 \arctan (5 x) + C)$

Step 8.2

Rewrite $\frac{1}{25} (5 \arctan (5 x) + C)$ as $\frac{1}{25} \cdot 5 \arctan (5 x) + C$ .

$\frac{1}{25} \cdot 5 \arctan (5 x) + C$

Step 8.3

Simplify.

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Step 8.3.1

Combine $\frac{1}{25}$ and $5$ .

$\frac{5}{25} \arctan (5 x) + C$

Step 8.3.2

Cancel the common factor of $5$ and $25$ .

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Step 8.3.2.1

Factor $5$ out of $5$ .

$\frac{5 (1)}{25} \arctan (5 x) + C$

Step 8.3.2.2

Cancel the common factors.

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Step 8.3.2.2.1

Factor $5$ out of $25$ .

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

Step 8.3.2.2.2

Cancel the common factor.

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

Step 8.3.2.2.3

Rewrite the expression.

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

Step 9

The answer is the antiderivative of the function $f (x) = \frac{1}{1 + 25 x^{2}}$ .

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