91Ó°ÊÓ

Determine which are probability density functions and justify your answer. $$f(x)=\frac{1}{2}(2-x) \text { over }[0,2]$$

Evaluate each integral using any algebraic method or trigonometric identity you think is appropriate, and then use a substitution to reduce it to a standard form. $$\int \frac{x^{2}}{x^{2}+1} d x$$

Evaluate the integrals. $$\int_{0}^{\pi} 3 \sin \frac{x}{3} d x$$

Evaluate the integrals without using tables. $$\int_{1}^{\infty} \frac{d x}{x^{1.001}}$$

Evaluate the integrals using integration by parts. $$\int \theta \cos \pi \theta d \theta$$

The instructions for the integrals have two parts, one for the Trapezoidal Rule and one for Simpson's Rule. I. Using the Trapezoidal Rule a. Estimate the integral with \(n=4\) steps and find an upper bound for \(\left|E_{T}\right|\) b. Evaluate the integral directly and find \(\left|E_{T}\right|\) c. Use the formula \(\left(\left|E_{T}\right| /(\text { true value })\right) \times 100\) to express \(\left|E_{T}\right|\) as a percentage of the integral's true value. II. Using Simpson's Rule a. Estimate the integral with \(n=4\) steps and find an upper bound for \(\left|E_{S}\right|\) b. Evaluate the integral directly and find \(\left|E_{S}\right|\) c. Use the formula \(\left(\left|E_{S}\right| /(\text { true value })\right) \times 100\) to express \(\left|E_{S}\right|\) as a percentage of the integral's true value. $$\int_{1}^{3}(2 x-1) d x$$

Expand the quotients by partial fractions. $$\frac{5 x-7}{x^{2}-3 x+2}$$

Evaluate each integral using any algebraic method or trigonometric identity you think is appropriate, and then use a substitution to reduce it to a standard form. $$\int(\sec x-\tan x)^{2} d x$$

Evaluate the integrals. $$\int \cos ^{3} x \sin x d x$$

Evaluate the integrals. \(\int_{-2}^{2} \frac{d x}{4+x^{2}}\)