91Ó°ÊÓ

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. $$ \int \frac{2 t+1}{t^{2}+t-4} d t $$

Determining Whether an Integral Is Improper In Exercises \(1-8\) , decide whether the integral is improper. Explain your reasoning. $$ \int_{0}^{\infty} \cos x d x $$

Use a table of integrals with forms involving the trigonometric functions to find the indefinite integral. \(\int \frac{\sin ^{4} \sqrt{x}}{\sqrt{x}} d x\)

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

Using Trigonometric Substitution In Exercises \(5-8,\) find the indefinite integral using the substitution \(x=4 \sin \theta .\) $$ \int \frac{4}{x^{2} \sqrt{16-x^{2}}} d x $$

Determining Whether an Integral Is Improper In Exercises \(1-8\) , decide whether the integral is improper. Explain your reasoning. $$ \int_{-\infty}^{\infty} \frac{\sin x}{4+x^{2}} d x $$

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

Using Trigonometric Substitution In Exercises \(5-8,\) find the indefinite integral using the substitution \(x=4 \sin \theta .\) $$ \int \frac{\sqrt{16-x^{2}}}{x} d x $$

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

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. $$ \int \frac{1}{\sqrt{x}(1-2 \sqrt{x})} d x $$