question\nevaluate the indefinite integral given below.\n int (-18x^{2}-10)csc^{2}(3x^{3}+5x)dx \nprovide…

Explanation:

Step1: Set substitution variable

Let $u = 3x^{3}+5x$. Then $du=(9x^{2} + 5)dx$, and $- 2du=(-18x^{2}-10)dx$.

Step2: Rewrite the integral

The integral $\int(-18x^{2}-10)\csc^{2}(3x^{3}+5x)dx$ becomes $\int - 2\csc^{2}(u)du$.

Step3: Integrate with respect to u

We know that $\int\csc^{2}(u)du=-\cot(u)+C$. So, $\int - 2\csc^{2}(u)du = 2\cot(u)+C$.

Step4: Substitute back u

Substituting $u = 3x^{3}+5x$ back, we get $2\cot(3x^{3}+5x)+C$.

Answer:

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