91Ó°ÊÓ

Mathematical Induction Use mathematical induction to verify that the following integral converges for any positive integer \(n .\) $$\int_{0}^{\infty} x^{n} e^{-x} d x$$

True or False? In Exercises \(51-54\) , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(x=\sin \theta,\) then $$ \int \frac{d x}{\sqrt{1-x^{2}}}=\int d \theta $$

Using Two Methods Evaluate $$ \int_{0}^{1} \frac{x}{1+x^{4}} d x $$ in two different ways, one of which is partial fractions.

Differential Equation In Exercises \(51-56,\) solve the differential equation. $$ \frac{d y}{d x}=\left(e^{x}+5\right)^{2} $$

In Exercises \(43-60,\) (a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using L'Hopital's Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). $$ \lim _{x \rightarrow 0^{+}}(1+x)^{1 / x} $$

Finding an Indefinite Integral In Exercises \(47-56\) , find the indefinite integral. Use a computer algebra system to confirm your result. $$ \int \frac{\cot ^{2} t}{\csc t} d t $$

In Exercises 49–54, use the tabular method to find the integral. $$ \int x^{3} \sin x d x $$

Differential Equation In Exercises \(51-56,\) solve the differential equation. $$ \frac{d y}{d x}=\left(4-e^{2 x}\right)^{2} $$

True or False? In Exercises \(51-54\) , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(x=\sec \theta,\) then $$ \int \frac{\sqrt{x^{2}-1}}{x} d x=\int \sec \theta \tan \theta d \theta $$

In Exercises 49–54, use the tabular method to find the integral. $$ \int x^{3} \cos 2 x d x $$