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

The integral of 1/(a^2x^2+b^2) is 1/(ab)arctan(a/b x)+C

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integral of 1/(a^2x^2+b^2)

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

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

Solution steps

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

Apply Integral Substitution

= \int \frac{1}{ab ( 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}{ab} \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}{ab} arctan (u)

Substitute back

u = \frac{a}{b} x

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

Add a constant to the solution

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

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