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Activity 5.3.3.
Evaluate each of the following indefinite integrals by using these steps:
-
Find two functions within the integrand that form (up to a possible missing constant) a function-derivative pair;
-
Make a substitution and convert the integral to one involving
\(u\) and
\(du\text{;}\)
-
Evaluate the new integral in
\(u\text{;}\)
-
Convert the resulting function of
\(u\) back to a function of
\(x\) by using your earlier substitution;
-
Check your work by differentiating the function of
\(x\text{.}\) You should come up with the integrand originally given.
(a)
\(\displaystyle \int \frac{x^2}{5x^3+1} \, dx\)
(b)
\(\displaystyle \int e^x \sin(e^x) \, dx\)
(c)
\(\displaystyle \int \frac{\cos(\sqrt{x})}{\sqrt{x}} \, dx\)
