Integration by parts

Evaluate each of the following indefinite integrals. Check each antiderivative that you find by differentiating.

1. $\displaystyle \int te^{-t} \, dt$

2. $\displaystyle \int 4x \sin(3x) \, dx$

3. $\displaystyle \int z \sec^2(z) \,dz$

   Note: after applying integration by parts, we need to evaluate $\int \tan(z) \, dz$. To do so, recall that $\tan(z) = \frac{\sin(z)}{\cos(z)}$ and think about a $u$-substitution that leads to an antiderivative for $\tan(z)$.

4. $\displaystyle \int x \ln(x) \, dx$

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)…

Evaluate each of the following indefinite integrals.

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

2. $\displaystyle \int t^3 \ln(t) \, dt$

3. $\displaystyle \int e^z \sin(z) \, dz$

4. $\displaystyle \int s^2 e^{3s} \, ds$

5. $\displaystyle \int t \arctan(t) \,dt$ (Hint: At a certain point in this problem, it is very helpful to note that $\frac{t^2}{1+t^2} = 1 - \frac{1}{1+t^2}$.)

Let $f(t) = te^{-2t}$ and $F(x) = \int_0^x f(t) \, dt$.

1. Determine $F'(x)$.

2. Use the First FTC to find a formula for $F$ that does not involve an integral.

3. Is $F$ an increasing or decreasing function for $x \gt 0$? Why?

Consider the indefinite integral given by $\int e^{2x} \cos(e^x) \, dx$.

1. Noting that $e^{2x} = e^x \cdot e^x$, use the substitution $z = e^{x}$ to determine a new, equivalent integral in the variable $z$.

2. Evaluate the integral you found in (a) using an appropriate technique.

3. How is the problem of evaluating $\int e^{2x} \cos(e^{2x}) \, dx$ different from evaluating the integral in (a)? Do so.

4. Evaluate each of the following integrals as well, keeping in mind the approach(es) used earlier in thi…

For each of the following indefinite integrals, determine whether you would use $u$-substitution, integration by parts, neither*, or both to evaluate the integral. In each case, write one sentence to explain your reasoning, and include a statement of any substitutions used. (That is, if you decide in a problem to let $u = e^{3x}$, you should state that, as well as that $du = 3e^{3x} \, dx$.) Finally, use your chosen approach to…