What is the integral of cot^7(x/2)csc^2(x/2) ?

The integral of cot^7(x/2)csc^2(x/2) is -1/4 cot^8(x/2)+C

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integral of cot^7(x/2)csc^2(x/2)

\int cot^{7} (\frac{x}{2}) csc^{2} (\frac{x}{2}) d x

- \frac{1}{4} cot^{8} (\frac{x}{2}) + C

Solution steps

\int cot^{7} (\frac{x}{2}) csc^{2} (\frac{x}{2}) d x

Apply u-substitution

: \int - 2 u^{7} d u

= \int - 2 u^{7} d u

Take the constant out:

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

= - 2 \cdot \int u^{7} d u

Apply the Power Rule

: \frac{u ^{8}}{8}

= - 2 \cdot \frac{u ^{8}}{8}

Substitute back

u = cot (\frac{x}{2})

= - 2 \cdot \frac{cot ^{8} ( \frac{x}{2} )}{8}

Simplify

- 2 \cdot \frac{cot ^{8} ( \frac{x}{2} )}{8} : - \frac{1}{4} cot^{8} (\frac{x}{2})

= - \frac{1}{4} cot^{8} (\frac{x}{2})

Add a constant to the solution

= - \frac{1}{4} cot^{8} (\frac{x}{2}) + C

(- 2, - 9), (0, - 3)

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

y^{^{'}} = \frac{3 x ^{2} - 1}{3 + 2 y}

\frac{d}{d x} (a x sin (k x) + b x cos (k x))

f (x) = 5 x + 2

What is the integral of cot^7(x/2)csc^2(x/2) ?
The integral of cot^7(x/2)csc^2(x/2) is -1/4 cot^8(x/2)+C