What is the integral of 1/((5+x^2)) ?

The integral of 1/((5+x^2)) is 1/(sqrt(5))arctan(x/(sqrt(5)))+C

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integral of 1/((5+x^2))

integral

\frac{1}{( 5 + x ^{2} )}

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

Solution steps

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

Apply Integral Substitution

= \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

= \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)

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

Substitute back

u = \frac{x}{5}

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

Add a constant to the solution

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

e^{2} x - 1

(\frac{1}{x - 1})^{\frac{- 1}{2}}

\frac{x}{( x ^{2} - 1 ) ^{\frac{1}{2}}}

\frac{1}{( x ^{3} + x ^{2} + x + 1 )}

(1 - x)^{2}

What is the integral of 1/((5+x^2)) ?
The integral of 1/((5+x^2)) is 1/(sqrt(5))arctan(x/(sqrt(5)))+C