91Ó°ÊÓ

Determine whether the given improper integral is convergent or divergent. If it converges, then evaluate it. \(\int_{0}^{1} \ln (x) d x\)

Decompose the given rational function into partial fractions. Calculate the coefficients. $$ \frac{2 x^{3}-4 x^{2}-13 x+76}{\left(x^{2}-4 x+4\right)\left(x^{2}+6 x+9\right)} $$

Explicitly calculate the partial fraction decomposition of the given rational function. \(\frac{3 x^{3}-5 x^{2}+10 x-19}{\left(x^{2}+4\right)\left(x^{2}+3\right)}\)

Determine whether the given improper integral converges or diverges. If it converges, then evaluate it. $$ \int_{0}^{\infty}(2 / 3)^{x} d x $$

Explicitly calculate the partial fraction decomposition of the given rational function. \(\frac{2 x^{4}+15 x^{2}+30}{\left(x^{2}+4\right)\left(x^{2}+3\right)^{2}}\)

Evaluate the given definite integral. $$ \int_{0}^{\pi / 2} \sqrt{\sin (x)} \cos ^{3}(x) d x $$

Evaluate the given integral by making a trigonometric substitution (even if you spot another way to evaluate the integral). $$ \int \frac{1}{x \sqrt{1+x^{2}}} d x $$

Integrate by parts to evaluate the given definite integral. $$ \int_{0}^{1} x 3^{x} d x $$

Determine whether the given improper integral is convergent or divergent. If it converges, then evaluate it. \(\int_{1}^{\sqrt{2}} \frac{1}{x \sqrt{x^{2}-1}} d x\)

Decompose the given rational function into partial fractions. Calculate the coefficients. $$ \frac{8}{(x-1)^{3}(x+1)} $$