What is the integral of (tan(3x)+cot(3x))^2 ?

The integral of (tan(3x)+cot(3x))^2 is 1/3 tan(3x)-1/3 cot(3x)+C

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integral of (tan(3x)+cot(3x))^2

\int (tan (3 x) + cot (3 x))^{2} d x

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

Solution steps

\int (tan (3 x) + cot (3 x))^{2} d x

Expand

(tan (3 x) + cot (3 x))^{2} : tan^{2} (3 x) + 2 tan (3 x) cot (3 x) + cot^{2} (3 x)

= \int tan^{2} (3 x) + 2 tan (3 x) cot (3 x) + cot^{2} (3 x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int tan^{2} (3 x) d x + \int 2 tan (3 x) cot (3 x) d x + \int cot^{2} (3 x) d x

\int tan^{2} (3 x) d x = - x + \frac{1}{3} tan (3 x)

\int 2 tan (3 x) cot (3 x) d x = 2 x

\int cot^{2} (3 x) d x = - x - \frac{1}{3} cot (3 x)

= - x + \frac{1}{3} tan (3 x) + 2 x - x - \frac{1}{3} cot (3 x)

Simplify

= \frac{1}{3} tan (3 x) - \frac{1}{3} cot (3 x)

Add a constant to the solution

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

n = 0 \sum \infty \frac{5 ^{n}}{3 ^{n}}

\frac{s}{s - 1}

\int (1 - 8 x) (ln (x) + 1) d x

6 x^{2} + x y + 6 y^{2} = 13, (1, 1)

What is the integral of (tan(3x)+cot(3x))^2 ?
The integral of (tan(3x)+cot(3x))^2 is 1/3 tan(3x)-1/3 cot(3x)+C