91Ó°ÊÓ

In Exercises \(21-32,\) express the integrand as a sum of partial fractions and evaluate the integrals. $$\int \frac{2 \theta^{3}+5 \theta^{2}+8 \theta+4}{\left(\theta^{2}+2 \theta+2\right)^{2}} d \theta$$

Use a substitution to change the integral into one you can find in the table. Then evaluate the integral. $$\int \frac{\sqrt{x}}{\sqrt{1-x}} d x$$

Evaluate the integrals. $$\int_{0}^{\pi / 2} \theta \sqrt{1-\cos 2 \theta} d \theta$$

Converge. Evaluate the integrals without using tables. $$\int_{0}^{2} \frac{d x}{\sqrt{|x-1|}}$$

The length of one arch of the curve \(y=\sin x\) is given by $$L=\int_{0}^{\pi} \sqrt{1+\cos ^{2} x} d x$$ Estimate \(L\) by Simpson's Rule with \(n=8.\)

Some integrals do not require integration by parts. $$\int \frac{\cos \sqrt{x}}{\sqrt{x}} d x$$

Use a substitution to change the integral into one you can find in the table. Then evaluate the integral. $$\int \frac{\sqrt{2-x}}{\sqrt{x}} d x$$

In Exercises \(21-32,\) express the integrand as a sum of partial fractions and evaluate the integrals. $$\int \frac{\theta^{4}-4 \theta^{3}+2 \theta^{2}-3 \theta+1}{\left(\theta^{2}+1\right)^{3}} d \theta$$

Use any method to evaluate the integrals. Most will require trigonometric substitutions, but some can be evaluated by other methods. $$\int \frac{x d x}{25+4 x^{2}}$$

Evaluate the integrals. $$\int_{-\pi}^{\pi}\left(1-\cos ^{2} t\right)^{3 / 2} d t$$