91Ó°ÊÓ

Use the approaches discussed in this section to evaluate the following integrals. $$\int_{0}^{1} \sqrt{1+\sqrt{x}} d x$$

Evaluate the following integrals. $$\int \frac{x^{2}+3 x+2}{x\left(x^{2}+2 x+2\right)} d x$$

Evaluate the following integrals. $$\int \frac{d x}{x^{3} \sqrt{x^{2}-1}}, x>1$$

Use a table of integrals to solve the following problems. Find the area of the region bounded by the graph of \(y=\frac{1}{\sqrt{x^{2}-2 x+2}}\) and the \(x\) -axis between \(x=0\) and \(x=3.\)

Use integration by parts to derive the following reduction formulas. $$\int x^{n} e^{a x} d x=\frac{x^{n} e^{a x}}{a}-\frac{n}{a} \int x^{n-1} e^{a x} d x, \quad \text { for } a \neq 0$$

Use the approaches discussed in this section to evaluate the following integrals. $$\int \sin x \sin 2 x d x$$

Evaluate the following integrals. $$\int \frac{2 x^{2}+5 x+5}{(x+1)\left(x^{2}+2 x+2\right)} d x$$

Determine whether the following statements are true and give an explanation or counterexample. a. If \(m\) is a positive integer, then \(\int_{0}^{\pi} \cos ^{2 m+1} x d x=0\) b. If \(m\) is a positive integer, then \(\int_{0}^{\pi} \sin ^{m} x d x=0\)

Evaluate the following integrals. $$\int \frac{d x}{x\left(x^{2}-1\right)^{3 / 2}}, x>1$$

Use integration by parts to derive the following reduction formulas. $$\int x^{n} \cos a x d x=\frac{x^{n} \sin a x}{a}-\frac{n}{a} \int x^{n-1} \sin a x d x, \quad \text { for } a \neq 0$$