What is the integral of 1/(81+36x^2) ?

The integral of 1/(81+36x^2) is 1/54 arctan((2x)/3)+C

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integral of 1/(81+36x^2)

\int \frac{1}{81 + 36 x ^{2}} d x

\frac{1}{54} arctan (\frac{2 x}{3}) + C

Solution steps

\int \frac{1}{81 + 36 x ^{2}} d x

Apply Integral Substitution

= \int \frac{1}{54 ( 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}{54} \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}{54} arctan (u)

Substitute back

u = \frac{6}{9} x

= \frac{1}{54} arctan (\frac{6}{9} x)

Simplify

\frac{1}{54} arctan (\frac{6}{9} x) : \frac{1}{54} arctan (\frac{2 x}{3})

= \frac{1}{54} arctan (\frac{2 x}{3})

Add a constant to the solution

= \frac{1}{54} arctan (\frac{2 x}{3}) + C

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What is the integral of 1/(81+36x^2) ?
The integral of 1/(81+36x^2) is 1/54 arctan((2x)/3)+C