Which of the following is the correct formula for integration by parts?

Evaluate the integral: $\int x\sin xdx$

Evaluate the integral: ${\int }_0^{\frac{\pi }{2}}\sin x\cdot \ln \left(\cos x\right)dx$

Which of the following is a type of partial fraction decomposition?

Which step is crucial when decomposing a rational function into partial fractions?

Evaluate the integral: $\int \frac{1}{x^2(x^2+4)}dx$

Which of the following integrals is improper?

Evaluate the improper integral.

${\int }_{-\infty }^{\infty }\frac{e^{4x}}{1+e^{8x}}dx$

Evaluate the improper integral ${\int }_1^{\infty }\frac{8{\tan }^{-1}x}{1+x^2}dx$.