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

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

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integral of 1/(ax^2+b)

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

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

Solution steps

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

Apply Integral Substitution

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

Substitute back

u = \frac{a}{b} x

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

Combine same powers :

\frac{x}{y} = \frac{x}{y}

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

Add a constant to the solution

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

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