§5.5 Other options for finding algebraic antiderivatives

Introduction

The Method of Partial Fractions

Using an Integral Table

Using Computer Algebra Systems

Summary

For each of the indefinite integrals below, the main question is to decide whether the integral can be evaluated using $u$-substitution, integration by parts, a combination of the two, or neither. For integrals for which your answer is affirmative, state the substitution(s) you would use. It is not necessary to actually evaluate any of the integrals completely, unless the integral can be evaluated immediately using a familiar basic antiderivative.

$\displaystyle \int x^2 \sin(x^3) \, dx$, $\displaystyle \int x^2 \sin(x) \, dx$, $\displaystyle \int \sin(x^3) \, dx$, $\displaystyle \int x^5 \sin(x^3) \, dx$

Evaluate

.

$\displaystyle \int \frac{1}{x^2 - 2x - 3} \, dx$, given that $\displaystyle \frac{1}{x^2 - 2x - 3} = \frac{1/4}{x-3} - \frac{1/4}{x+1}$

$\displaystyle \int \frac{x^2+1}{x^3 - x^2} \, dx$, given that $\displaystyle \frac{x^2+1}{x^3 - x^2} = -\frac{1}{x} - \frac{1}{x^2} + \frac{2}{x-1}$

$\displaystyle \int \frac{x-2}{x^4 + x^2}\, dx$, given that $\displaystyle \frac{x-2}{x^4 + x^2} = \frac{1}{x} - \frac{2}{x^2} + \frac{-x+2}{1+x^2}$

Evaluate the integral

.

$\displaystyle \int \sqrt{x^2 + 4} \, dx$

$\displaystyle \int \frac{x}{\sqrt{x^2 +4}} \, dx$

$\displaystyle \int \frac{2}{\sqrt{16+25x^2}}\, dx$

$\displaystyle \int \frac{1}{x^2 \sqrt{49-36x^2}} \, dx$

For each of the following integrals involving rational functions, (1) use a CAS to find the partial fraction decomposition of the integrand; (2) evaluate the integral of the resulting function without the assistance of technology; (3) use a CAS to evaluate the original integral to test and compare your result in (2). -$\int \frac{x^3 + x + 1}{x^4 - 1} \, dx$-$\int \frac{x^5 + x^2 + 3}{x^3 - 6x^2 + 11x - 6} \, dx$-$\int \frac{x^2 - x - 1}{(x-3)^3} \, dx$

For each of the following integrals involving radical functions, (1) use an appropriate $u$-substitution along with Appendix to evaluate the integral without the assistance of technology, and (2) use a CAS to evaluate the original integral to test and compare your result in (1). -$\int \frac{1}{x \sqrt{9x^2 + 25}} \, dx$-$\int x \sqrt{1 + x^4} \, dx$-$\int e^x \sqrt{4 + e^{2x}} \, dx$-$\int \frac{\tan(x)}{\sqrt{9 - \cos^2(x)}} \, dx$

Consider the indefinite integral given by

. - Explain why $u$-substitution does not offer a way to simplify this integral by discussing at least two different options you might try for $u$. - Explain why integration by parts does not seem to be a reasonable way to proceed, either, by considering one option for $u$ and $dv$. - Is there any line in the integral table in Appendix that is helpful for this integral? - Evaluate the given integral using WolframAlpha. What do you observe?