Question

1. $\int - 40 x \csc ^ { 2 } ( 5 x ^ { 2 } + 4 ) d x$

Explanation:

Step1: Let \( u = 5x^{2}+4 \)

Differentiate \( u \) with respect to \( x \): \( du=10x dx\), so \( - 40x dx=-4du \)

Step2: Substitute into the integral

The integral \( \int - 40x\csc^{2}(5x^{2}+4)dx \) becomes \( \int - 4\csc^{2}(u)du \)

Step3: Integrate \( \csc^{2}(u) \)

We know that \( \int\csc^{2}(u)du=-\cot(u)+C \). So \( \int - 4\csc^{2}(u)du = 4\cot(u)+C \)

Step4: Substitute back \( u = 5x^{2}+4 \)

The result is \( 4\cot(5x^{2}+4)+C \)

Answer:

\( 4\cot(5x^{2}+4)+C \)