What is the integral of (-18x^2-10)csc^2(3x^3+5x) ?

The integral of (-18x^2-10)csc^2(3x^3+5x) is 2cot(3x^3+5x)+C

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integral of (-18x^2-10)csc^2(3x^3+5x)

\int (- 18 x^{2} - 10) csc^{2} (3 x^{3} + 5 x) d x

2 cot (3 x^{3} + 5 x) + C

Solution steps

\int (- 18 x^{2} - 10) csc^{2} (3 x^{3} + 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 = 3 x^{3} + 5 x

= - 2 (- cot (3 x^{3} + 5 x))

Simplify

= 2 cot (3 x^{3} + 5 x)

Add a constant to the solution

= 2 cot (3 x^{3} + 5 x) + C

What is the integral of (-18x^2-10)csc^2(3x^3+5x) ?
The integral of (-18x^2-10)csc^2(3x^3+5x) is 2cot(3x^3+5x)+C