What is the integral of cot^4(5x) ?

The integral of cot^4(5x) is -1/15 cot^3(5x)+x+1/5 cot(5x)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of cot^4(5x)

\int cot^{4} (5 x) d x

- \frac{1}{15} cot^{3} (5 x) + x + \frac{1}{5} cot (5 x) + C

Solution steps

\int cot^{4} (5 x) d x

Apply u-substitution

= \int cot^{4} (u) \frac{1}{5} d u

Take the constant out:

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

= \frac{1}{5} \cdot \int cot^{4} (u) d u

Apply Integral Reduction

= \frac{1}{5} (- \frac{cot ^{3} ( u )}{3} - \int cot^{2} (u) d u)

\int cot^{2} (u) d u = - u - cot (u)

= \frac{1}{5} (- \frac{cot ^{3} ( u )}{3} - (- u - cot (u)))

Substitute back

u = 5 x

= \frac{1}{5} (- \frac{cot ^{3} ( 5 x )}{3} - (- 5 x - cot (5 x)))

Simplify

\frac{1}{5} (- \frac{cot ^{3} ( 5 x )}{3} - (- 5 x - cot (5 x))) : - \frac{1}{15} cot^{3} (5 x) + x + \frac{1}{5} cot (5 x)

= - \frac{1}{15} cot^{3} (5 x) + x + \frac{1}{5} cot (5 x)

Add a constant to the solution

= - \frac{1}{15} cot^{3} (5 x) + x + \frac{1}{5} cot (5 x) + C

What is the integral of cot^4(5x) ?
The integral of cot^4(5x) is -1/15 cot^3(5x)+x+1/5 cot(5x)+C