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

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

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integral of 1/(ax^2+1)

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

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

Solution steps

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

Apply Integral Substitution

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

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

u = a x

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

Add a constant to the solution

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

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