Other options for finding algebraic antiderivatives

Partial fractions for rational integrands, historical integral tables and how to match an entry, and what a computer algebra system adds.

Active Calculus, Other options for finding algebraic antiderivatives

Status

FlagsLoading your progress…

5 exercises

Activity 5.5.2

For each of the following problems, evaluate the integral by using the partial fraction decomposition provided.

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

2.5

Activity 5.5.3

For each of the following integrals, evaluate the integral using $u$ -substitution and/or an entry from the table found in Appendix.

$\displaystyle \int \sqrt{x^2 + 4} \, dx$
$\displaystyle \int \frac{x}{\sqrt{x^2 +4}} \, dx$
$\displaystyle \int \frac{2}{\sqrt{16+25x^2}}\, dx$
$\displaystyle \int \frac{1}{x^2 \sqrt{49-36x^2}} \, dx$

2.5

Exercise 6

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).

$\int \frac{x^3 + x + 1}{x^4 - 1} \, dx$
$\int \frac{x^5 + x^2 + 3}{x^3 - 6x^2 + 11x - 6} \, dx$
$\int \frac{x^2 - x - 1}{(x-3)^3} \, dx$

2.5

Exercise 7

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).

$\int \frac{1}{x \sqrt{9x^2 + 25}} \, dx$
$\int x \sqrt{1 + x^4} \, dx$
$\int e^x \sqrt{4 + e^{2x}} \, dx$
$\int \frac{\tan(x)}{\sqrt{9 - \cos^2(x)}} \, dx$

2.5

Exercise 8

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?
Eval…

2.5