91Ó°ÊÓ

Use any method to evaluate the integrals in Exercises \(15-34 .\) Most will require trigonometric substitutions, but some can be evaluated by other methods. $$ \int \frac{x d x}{25+4 x^{2}} $$

The integrals in Exercises \(1-34\) converge. Evaluate the integrals without using tables. $$\int_{0}^{2} \frac{d x}{\sqrt{|x-1|}}$$

In Exercises \(27-40\) , use a substitution to change the integral into one you can find in the table. Then evaluate the integral. $$ \int \frac{\sqrt{2-x}}{\sqrt{x}} d x $$

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

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

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 \(n=8\)

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

The integrals in Exercises \(1-44\) are in no particular order. Evaluate each integral using any algebraic method or trigonometric identity you think is appropriate. When necessary, use a substitution to reduce it to a standard form. $$ \int_{-1}^{1} \sqrt{1+x^{2}} \sin x d x $$

The integrals in Exercises \(1-44\) are in no particular order. Evaluate each integral using any algebraic method or trigonometric identity you think is appropriate. When necessary, use a substitution to reduce it to a standard form. $$ \int_{-1}^{0} \sqrt{\frac{1+y}{1-y}} d y $$

Evaluate the integrals. Some integrals do not require integration by parts. $$ \int x(\ln x)^{2} d x $$