What is the integral of (x^4csc^2(x^5))/(cot(x^5)) ?

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

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integral of (x^4csc^2(x^5))/(cot(x^5))

\int \frac{x ^{4} csc ^{2} ( x ^{5} )}{cot ( x ^{5} )} d x

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

Solution steps

\int \frac{x ^{4} csc ^{2} ( x ^{5} )}{cot ( x ^{5} )} d x

Apply u-substitution

= \int - \frac{1}{5 u} 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 \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - \frac{1}{5} ln ∣ u ∣

Substitute back

u = cot (x^{5})

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

Add a constant to the solution

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

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