Week 11 Worksheet

Compute the following antiderivatives.

1. \(\int \frac{7}{z} dz\)
2. \(\int \frac{x^3 + x + 1}{x^2} dx\)
3. \(\int \frac{10x}{3 + 5x^2} dx\)
4. \(\int p(p+2)^6dp\)
5. \(\int \frac{(\log_2 (5x + 1))^2}{5x + 1} dx\)
6. \(\int \frac{10^{5\sqrt x + 2}}{\sqrt x} dx\)
7. \(\int \frac{\sec x + \cos x}{\cos x} dx\) (Hint: recall that \((\tan x)' = \sec^2x\))
8. Challenge: It is a theorem that if \(f\) is a function whose domain is an interval, and \(f\) has an antiderivative, say \(F\), then all antiderivatives of \(f\) are of the form \(F(x) + C\) for constants \(C\). Give an example to show this need not be true if the domain of \(f\) is not an interval, e.g. if \(f\) is a function defined only on \((-\infty, 0) \cup (0, \infty)\).