What is the integral of (sec^2(x))/(9+tan^2(x)) ?

The integral of (sec^2(x))/(9+tan^2(x)) is 1/3 arctan((tan(x))/3)+C

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integral of (sec^2(x))/(9+tan^2(x))

\int \frac{sec ^{2} ( x )}{9 + tan ^{2} ( x )} d x

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

Solution steps

\int \frac{sec ^{2} ( x )}{9 + tan ^{2} ( x )} d x

Apply u-substitution

: \int \frac{1}{9 + u ^{2}} d u

= \int \frac{1}{9 + u ^{2}} d u

Apply Integral Substitution

= \int \frac{1}{3 ( v ^{2} + 1 )} d v

Take the constant out:

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

= \frac{1}{3} \cdot \int \frac{1}{v ^{2} + 1} d v

Use the common integral:

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

= \frac{1}{3} arctan (v)

Substitute back

= \frac{1}{3} arctan (\frac{tan ( x )}{3})

Add a constant to the solution

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

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What is the integral of (sec^2(x))/(9+tan^2(x)) ?
The integral of (sec^2(x))/(9+tan^2(x)) is 1/3 arctan((tan(x))/3)+C