Chapter 6: Problem 2
Decide whether the integral is improper. Explain your reasoning. $$ \int_{1}^{3} \frac{d x}{x^{2}} $$
Over 30 million students worldwide already upgrade their learning with Vaia!
All the tools & learning materials you need for study success - in one app.
Get started for free
Find the integral. Use a computer algebra system to confirm your result. $$ \int \tan ^{4} \frac{x}{2} \sec ^{4} \frac{x}{2} d x $$
Think About It In Exercises 55-58, L'Hopital's Rule is used incorrectly. Describe the error. \(\lim _{x \rightarrow 0} \frac{e^{2 x}-1}{e^{x}}=\lim _{x \rightarrow 0} \frac{2 e^{2 x}}{e^{x}}=\lim _{x \rightarrow 0} 2 e^{x}=2\)
Evaluate \(\lim _{x \rightarrow \infty}\left[\frac{1}{x} \cdot \frac{a^{x}-1}{a-1}\right]^{1 / x}\) where \(a>0, \quad a \neq 1\).
Think About It In Exercises 55-58, L'Hopital's Rule is used incorrectly. Describe the error.\(\begin{aligned} \lim _{x \rightarrow \infty} \operatorname{xec} \operatorname{sen} \frac{1}{x} &=\lim _{x \rightarrow \infty} \frac{\cos (1 / x)}{1 / x} \\ &=\lim _{x \rightarrow \infty} \frac{-\sin (1 / x)]\left(1 / x^{2}\right)}{-1 \times x^{2}} \\ &=0 \end{aligned}\)
Find the area of the region bounded by the graphs of the equations.$$ y=\sin x, \quad y=\sin ^{3} x, \quad x=0, \quad x=\pi / 2 $$
What do you think about this solution?
We value your feedback to improve our textbook solutions.
