What is the integral of cot^3(x)csc^5(x) ?

The integral of cot^3(x)csc^5(x) is -1/7 csc^7(x)+1/5 csc^5(x)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of cot^3(x)csc^5(x)

\int cot^{3} (x) csc^{5} (x) d x

- \frac{1}{7} csc^{7} (x) + \frac{1}{5} csc^{5} (x) + C

Solution steps

\int cot^{3} (x) csc^{5} (x) d x

Apply u-substitution

: \int - u^{4} (u^{2} - 1) d u

= \int - u^{4} (u^{2} - 1) d u

Take the constant out:

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

= - \int u^{4} (u^{2} - 1) d u

Expand

u^{4} (u^{2} - 1) : u^{6} - u^{4}

= - \int u^{6} - u^{4} d u

Apply the Sum Rule:

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

= - (\int u^{6} d u - \int u^{4} d u)

\int u^{6} d u = \frac{u ^{7}}{7}

\int u^{4} d u = \frac{u ^{5}}{5}

= - (\frac{u ^{7}}{7} - \frac{u ^{5}}{5})

Substitute back

u = csc (x)

= - (\frac{csc ^{7} ( x )}{7} - \frac{csc ^{5} ( x )}{5})

Simplify

- (\frac{csc ^{7} ( x )}{7} - \frac{csc ^{5} ( x )}{5}) : - \frac{1}{7} csc^{7} (x) + \frac{1}{5} csc^{5} (x)

= - \frac{1}{7} csc^{7} (x) + \frac{1}{5} csc^{5} (x)

Add a constant to the solution

= - \frac{1}{7} csc^{7} (x) + \frac{1}{5} csc^{5} (x) + C

\int sin (2 x) cos^{5} (2 x) d x

\frac{\partial}{\partial x} (x^{4} sin (y^{9} z^{9}))

x \to 5 lim (\frac{x ^{2} - 25}{3 x - 15})

(1 - x)^{2} + x^{3}

What is the integral of cot^3(x)csc^5(x) ?
The integral of cot^3(x)csc^5(x) is -1/7 csc^7(x)+1/5 csc^5(x)+C