What is the integral of 4/(5+x^2) ?

The integral of 4/(5+x^2) is 4/(sqrt(5))arctan(x/(sqrt(5)))+C

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integral of 4/(5+x^2)

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

\frac{4}{5} arctan (\frac{x}{5}) + C

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Apply Integral Substitution

= 4 \cdot \int \frac{1}{5 ( 1 + u ^{2} )} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 4 \cdot \frac{1}{5} \cdot \int \frac{1}{1 + u ^{2}} d u

Use the common integral:

\int \frac{1}{1 + u ^{2}} d u = arctan (u)

= 4 \cdot \frac{1}{5} arctan (u)

Substitute back

u = \frac{x}{5}

= 4 \cdot \frac{1}{5} arctan (\frac{x}{5})

Simplify

4 \cdot \frac{1}{5} arctan (\frac{x}{5}) : \frac{4}{5} arctan (\frac{x}{5})

= \frac{4}{5} arctan (\frac{x}{5})

Add a constant to the solution

= \frac{4}{5} arctan (\frac{x}{5}) + C

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What is the integral of 4/(5+x^2) ?
The integral of 4/(5+x^2) is 4/(sqrt(5))arctan(x/(sqrt(5)))+C