91Ó°ÊÓ

(a) Evaluate the integral \(\int(5 x-1)^{2} d x\) by two methods: first square and integrate, then let \(u=5 x-1\) (b) Explain why the two apparently different answers obtained in part (a) are really equivalent.

Use a trigonometric identity to evaluate the integral. $$ \int \tan ^{2} x d x $$

Solve the initial-value problems. $$ \frac{d y}{d x}=\sqrt{5 x+1}, y(3)=-2 $$

Use a trigonometric identity to evaluate the integral. $$ \int \cot ^{2} x d x $$

Solve the initial-value problems. $$ \frac{d y}{d x}=2+\sin 3 x, y(\pi / 3)=0 $$

A traffic engineer monitors the rate at which cars enter the main highway during the afternoon rush hour. From her data she estimates that between 4: 30 P.M. and 5: 30 P. M. the rate \(R(t)\) at which cars enter the highway is given by the formula \(R(t)=100\left(1-0.0001 t^{2}\right)\) cars per minute, where \(t\) is the time (in minutes) since 4: 30 P.M. (a) When does the peak traffic flow into the highway occur? (b) Estimate the number of cars that enter the highway during the rush hour.

Solve the initial-value problems. $$ \frac{d y}{d t}=-e^{2 t}, y(0)=6 $$

Evaluate each limit by interpreting it as a Riemann sum in which the given interval is divided into \(n\) subintervals of equal width. $$ \lim _{n \rightarrow+\infty} \sum_{k=1}^{n} \frac{\pi}{4 n} \sec ^{2}\left(\frac{\pi k}{4 n}\right) ;\left[0, \frac{\pi}{4}\right] $$

Use the identities \(\cos 2 \theta=1-2 \sin ^{2} \theta=2 \cos ^{2} \theta-1\) to help evaluate the integrals $$ \text { (a) } \int \sin ^{2}(x / 2) d x \quad \text { (b) } \int \cos ^{2}(x / 2) d x $$

Evaluate each limit by interpreting it as a Riemann sum in which the given interval is divided into \(n\) subintervals of equal width. $$ \lim _{n \rightarrow+\infty} \sum_{k=1}^{n} \frac{n}{n^{2}+k^{2}} ;[0,1] $$