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).

1. $\int \frac{1}{x \sqrt{9x^2 + 25}} \, dx$

2. $\int x \sqrt{1 + x^4} \, dx$

3. $\int e^x \sqrt{4 + e^{2x}} \, dx$

4. $\int \frac{\tan(x)}{\sqrt{9 - \cos^2(x)}} \, dx$