What is the integral of csc^2(x)cot^5(x) ?

The integral of csc^2(x)cot^5(x) is -(cot^6(x))/6+C

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integral of csc^2(x)cot^5(x)

\int csc^{2} (x) cot^{5} (x) d x

- \frac{cot ^{6} ( x )}{6} + C

Solution steps

\int csc^{2} (x) cot^{5} (x) d x

Rewrite using trig identities

= \int \frac{cos ^{5} ( x )}{sin ^{7} ( x )} d x

Apply Trigonometric Substitution

: \int - u^{5} d u

= \int - u^{5} d u

Take the constant out:

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

= - \int u^{5} d u

Apply the Power Rule

: \frac{u ^{6}}{6}

= - \frac{u ^{6}}{6}

Substitute back

u = cot (x)

= - \frac{cot ^{6} ( x )}{6}

Add a constant to the solution

= - \frac{cot ^{6} ( x )}{6} + C

\int e^{\frac{1}{50} x} d x

\frac{d y}{d x} = - 5 y x

\int_{0}^{4} 1 + x d x

\int \frac{- 1}{1 - x} d x

\int_{1}^{3} (- x^{2} + 4 x - 3) d x

What is the integral of csc^2(x)cot^5(x) ?
The integral of csc^2(x)cot^5(x) is -(cot^6(x))/6+C