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 $du$;

• Evaluate the new integral in $u$;

• 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 $x$. You should come up with the integrand originally given.

1. $\displaystyle \int \frac{x^2}{5x^3+1} \, dx$

2. $\displaystyle \int e^x \sin(e^x) \, dx$

3. $\displaystyle \int \frac{\cos(\sqrt{x})}{\sqrt{x}} \, dx$