91Ó°ÊÓ

Compute the mean and median for a random variable with the probability density functions. \(f(x)=\frac{1}{8} x\) over \([0,4]\)

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. $$ \int \frac{d y}{\sqrt{e^{2 y}-1}} $$

Evaluate the integrals. \(\int_{0}^{\pi} \sqrt{1-\sin ^{2} t} d t\)

Evaluate the integrals by using a substitution prior to integration by parts. $$ \int e^{\sqrt{3 s+9}} d s $$

Use the table of integrals at the back of the book to evaluate the integrals in Exercises \(1-26 .\) $$ \int \cos \frac{\theta}{3} \cos \frac{\theta}{4} d \theta $$

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

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

Wing design The design of a new airplane requires a gasoline tank of constant cross-sectional area in each wing. A scale drawing of a cross-section is shown here. The tank must hold 5000 lb of gasoline, which has a density of 42 \(\mathrm{lb} / \mathrm{ft}^{3}\) . Estimate the length of the tank by Simpson's Rule. $$y_{0}=1.5 \mathrm{ft}, \quad y_{1}=1.6 \mathrm{ft}, \quad y_{2}=1.8 \mathrm{ft}, \quad y_{3}=1.9 \mathrm{ft}$$ $$y_{4}=2.0 \mathrm{ft}, \quad y_{5}=y_{6}=2.1 \mathrm{ft} \quad\( Horizontal spacing \)=1 \mathrm{ft}$$

Use any method to evaluate the integrals in Exercises \(15-34 .\) Most will require trigonometric substitutions, but some can be evaluated by other methods. $$ \int \frac{d x}{\left(x^{2}-1\right)^{3 / 2}}, \quad x>1 $$

Use the table of integrals at the back of the book to evaluate the integrals in Exercises \(1-26 .\) $$ \int \cos \frac{\theta}{2} \cos 7 \theta d \theta $$