What is the integral of (10-20x^4)csc^2(2x^5-5x) ?

The integral of (10-20x^4)csc^2(2x^5-5x) is 2cot(2x^5-5x)+C

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integral of (10-20x^4)csc^2(2x^5-5x)

\int (10 - 20 x^{4}) csc^{2} (2 x^{5} - 5 x) d x

2 cot (2 x^{5} - 5 x) + C

Solution steps

\int (10 - 20 x^{4}) csc^{2} (2 x^{5} - 5 x) d x

Apply u-substitution

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

Take the constant out:

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

= - 2 \cdot \int csc^{2} (u) d u

Use the common integral:

\int csc^{2} (u) d u = - cot (u)

= - 2 (- cot (u))

Substitute back

u = 2 x^{5} - 5 x

= - 2 (- cot (2 x^{5} - 5 x))

Simplify

= 2 cot (2 x^{5} - 5 x)

Add a constant to the solution

= 2 cot (2 x^{5} - 5 x) + C

What is the integral of (10-20x^4)csc^2(2x^5-5x) ?
The integral of (10-20x^4)csc^2(2x^5-5x) is 2cot(2x^5-5x)+C