What is the integral of (1+cot(x))^4(csc(x))^2 ?

The integral of (1+cot(x))^4(csc(x))^2 is -((1+cot(x))^5)/5+C

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integral of (1+cot(x))^4(csc(x))^2

\int (1 + cot (x))^{4} (csc (x))^{2} d x

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

Solution steps

\int (1 + cot (x))^{4} (csc (x))^{2} d x

Simplify

= \int csc^{2} (x) (1 + cot (x))^{4} d x

Apply u-substitution

: \int - u^{4} d u

= \int - u^{4} d u

Take the constant out:

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

= - \int u^{4} d u

Apply the Power Rule

: \frac{u ^{5}}{5}

= - \frac{u ^{5}}{5}

Substitute back

u = 1 + cot (x)

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

Add a constant to the solution

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

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What is the integral of (1+cot(x))^4(csc(x))^2 ?
The integral of (1+cot(x))^4(csc(x))^2 is -((1+cot(x))^5)/5+C