Activity 5.4.3.
Evaluate each of the following indefinite integrals, using the provided hints.
(a)
Evaluate \(\int \arctan(x) \, dx\) by using integration by parts with the substitution \(u = \arctan(x)\) and \(dv = 1 \, dx\text{.}\)
(b)
Evaluate \(\int \ln(z) \,dz\text{.}\) Consider a similar substitution to the one in (a).
(c)
Use the substitution \(z = t^2\) to transform the integral \(\int t^3 \sin(t^2) \, dt\) to a new integral in the variable \(z\text{,}\) and evaluate that new integral by parts.
(d)
Evaluate \(\int s^5 e^{s^3} \, ds\) using an approach similar to that described in (c).
(e)
Evaluate \(\int e^{2t} \cos(e^t) \, dt\text{.}\) You will find it helpful to note that \(e^{2t} = e^t \cdot e^t\text{.}\)
