Activity 5.4.3

Evaluate each of the following indefinite integrals, using the provided hints.

1. Evaluate $\int \arctan(x) \, dx$ by using integration by parts with the substitution $u = \arctan(x)$ and $dv = 1 \, dx$.

2. Evaluate $\int \ln(z) \,dz$. Consider a similar substitution to the one in (a).

3. Use the substitution $z = t^2$ to transform the integral $\int t^3 \sin(t^2) \, dt$ to a new integral in the variable $z$, and evaluate that new integral by parts.

4. Evaluate $\int s^5 e^{s^3} \, ds$ using an approach similar to that described in (c).

5. Evaluate $\int e^{2t} \cos(e^t) \, dt$. You will find it helpful to note that $e^{2t} = e^t \cdot e^t$.