What is the integral of (3sec^2(x))/(6+3tan(x)) ?

The integral of (3sec^2(x))/(6+3tan(x)) is ln|6+3tan(x)|+C

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integral of (3sec^2(x))/(6+3tan(x))

\int \frac{3 sec ^{2} ( x )}{6 + 3 tan ( x )} d x

ln ∣ 6 + 3 tan (x) ∣ + C

Solution steps

\int \frac{3 sec ^{2} ( x )}{6 + 3 tan ( x )} d x

Take the constant out:

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

= 3 \cdot \int \frac{sec ^{2} ( x )}{6 + 3 tan ( x )} d x

Apply u-substitution

: \int \frac{1}{3 u} d u

= 3 \cdot \int \frac{1}{3 u} d u

Take the constant out:

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

= 3 \cdot \frac{1}{3} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 3 \cdot \frac{1}{3} ln ∣ u ∣

Substitute back

u = 6 + 3 tan (x)

= 3 \cdot \frac{1}{3} ln ∣ 6 + 3 tan (x) ∣

Simplify

3 \cdot \frac{1}{3} ln ∣ 6 + 3 tan (x) ∣ : ln ∣ 6 + 3 tan (x) ∣

= ln ∣ 6 + 3 tan (x) ∣

Add a constant to the solution

= ln ∣ 6 + 3 tan (x) ∣ + C

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What is the integral of (3sec^2(x))/(6+3tan(x)) ?
The integral of (3sec^2(x))/(6+3tan(x)) is ln|6+3tan(x)|+C