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

The integral of (sec^2(6x))/(tan(6x)+2) is 1/6 ln|tan(6x)+2|+C

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

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

\frac{1}{6} ln ∣ tan (6 x) + 2 ∣ + C

Solution steps

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

Apply u-substitution

: \int \frac{1}{6 u} d u

= \int \frac{1}{6 u} d u

Take the constant out:

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

= \frac{1}{6} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{6} ln ∣ u ∣

Substitute back

u = tan (6 x) + 2

= \frac{1}{6} ln ∣ tan (6 x) + 2 ∣

Add a constant to the solution

= \frac{1}{6} ln ∣ tan (6 x) + 2 ∣ + C

\frac{3 s}{s ^{2} + 3 s + 16}

\frac{d}{d x} (x^{3} tan (x))

\int \frac{1}{5 x ( 2 + x )} d x

\frac{\partial}{\partial x} (e^{x + y + 4})

x \to 5 lim (x - 5)

What is the integral of (sec^2(6x))/(tan(6x)+2) ?
The integral of (sec^2(6x))/(tan(6x)+2) is 1/6 ln|tan(6x)+2|+C