91Ó°ÊÓ

Using Integration by Parts In Exercises \(7-10\) , evaluate the integral using integration by parts with the given choices of \(u\) and \(d v .\) $$ \int(4 x+7) e^{x} d x ; u=4 x+7, d v=e^{x} d x $$

Evaluating an Improper Integral In Exercises \(9-12\) , explain why the integral is improper and determine whether it diverges or converges. Evaluate the integral if it converges. $$ \int_{0}^{4} \frac{1}{\sqrt{x}} d x $$

Using Two Methods In Exercises \(5-10\) , evaluate the limit (a) using techniques from Chapters 1 and 3 and (b) using L'Hopital's Rule. $$ \lim _{x \rightarrow \infty} \frac{5 x^{2}-3 x+1}{3 x^{2}-5} $$

Choosing a Formula In Exercises \(5-14\) , select the basic integration formula you can use to find the integral, and identify \(u\) and \(a\) when appropriate. $$ \int \frac{3}{\sqrt{1-t^{2}}} d t $$

Finding an Indefinite Integral Involving Sine and cosine In Exercises \(1-12,\) find the indefinite integral. $$ \int \cos ^{2} 3 x d x $$

Using Partial Fractions In Exercises \(5-22,\) use partial fractions to find the indefinite integral. $$ \int \frac{x^{2}+12 x+12}{x^{3}-4 x} d x $$

Using Trigonometric Substitution In Exercises \(9-12\) , find the indefinite integral using the substitution \(x=5\) sec \(\theta\) . $$ \int \frac{1}{\sqrt{x^{2}-25}} d x $$

Using Integration by Parts In Exercises \(7-10\) , evaluate the integral using integration by parts with the given choices of \(u\) and \(d v .\) $$ \int x \sin 3 x d x ; u=x, d v=\sin 3 x d x $$

Use a table of integrals with forms involving \(e^{u}\) to find the indefinite integral. \(\int \frac{1}{1+e^{2 x}} d x\)

Choosing a Formula In Exercises \(5-14\) , select the basic integration formula you can use to find the integral, and identify \(u\) and \(a\) when appropriate. $$ \int \frac{-2 x}{\sqrt{x^{2}-4}} d x $$