91Ó°ÊÓ

Evaluate the integrals using appropriate substitutions. $$ \int \sec ^{3} 2 x \tan 2 x d x $$

Writing Make a list of important features of a velocity versus time curve, and interpret each feature in terms of the motion.

Evaluate the integrals by any method. $$ \int_{0}^{1 / \sqrt{3}} \frac{1}{1+9 x^{2}} d x $$

A particle moves along an \(s\) -axis with position function \(s=s(t)\) and velocity function \(v(t)=s^{\prime}(t) .\) Use the given information to find \(s(t)\). $$ v(t)=32 t ; \quad s(0)=20 $$

Sketch the region described and find its area. The region under the curve \(y=3 \sin x\) and over the interval \([0,2 \pi / 3] .\)

Sketch the region described and find its area. The region below the interval \([-2,-1]\) and above the curve \(y=x^{3} .\)

Writing Use Riemann sums to argue informally that integrating speed over a time interval produces the distance traveled.

(a) Give a geometric argument to show that $$ \frac{1}{x+1}<\int_{x}^{x+1} \frac{1}{t} d t<\frac{1}{x}, \quad x>0 $$ (b) Use the result in part (a) to prove that \(\frac{1}{x+1}<\ln \left(1+\frac{1}{x}\right)<\frac{1}{x}, \quad x>0\) (c) Use the result in part (b) to prove that $$ e^{x /(x+1)}<\left(1+\frac{1}{x}\right)^{x}0 $$ and hence that $$ \lim _{x \rightarrow+\infty}\left(1+\frac{1}{x}\right)^{x}=e $$ (d) Use the result in part (b) to prove that $$ \left(1+\frac{1}{x}\right)^{x}0 $$

Evaluate the integrals by any method. $$ \int_{1}^{\sqrt{2}} \frac{x}{3+x^{4}} d x $$

A particle moves along an \(s\) -axis with position function \(s=s(t)\) and velocity function \(v(t)=s^{\prime}(t) .\) Use the given information to find \(s(t)\). $$ v(t)=\cos t ; \quad s(0)=2 $$