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

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

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

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

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

Solution steps

\int \frac{1}{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 = 5 x

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

Add a constant to the solution

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

d er i v a t i v e 3 - 5 x

d er i v a t i v e f (x) = a x^{2} + b x

\int \frac{2}{x ^{0.85}} d x

\frac{\partial}{\partial x} (2 x y + x f (x, y) (\frac{y}{x}))

\frac{\partial}{\partial y} (3 + 2 x y)

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