§5.4 Integration by parts

Introduction

Reversing the Product Rule: Integration by Parts

Some Subtleties with Integration by Parts

In Section 2.3, we developed the Product Rule and studied how it is employed to differentiate a product of two functions. In particular, recall that if $f$ and $g$ are differentiable functions of $x$, then

.

For each of the following functions, use the Product Rule to find the function's derivative. Be sure to label each derivative by name (e.g., the derivative of $g(x)$ should be labeled $g'(x)$).

-$g(x) = x\sin(x)$-$h(x) = xe^x$-$p(x) = x\ln(x)$-$q(x) = x^2 \cos(x)$-$r(x) = e^x \sin(x)$

Evaluate the indefinite integral

using integration by parts.

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

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

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

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 $\tan(z)$.

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

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

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

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.

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

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