Overview

Sometimes we need to find integrals of functions which are products of two different functions. For example, take your time and think how one can integrate $\displaystyle\int x\cos x\,dx$. Or, try solving $\displaystyle\int x\sec^2 x\,dx$.

As you can see, these are integrals of products of two completely different functions — $x$ and $\cos x$, or $x$ and $\sec^2 x$. In this chapter, you will learn how to integrate these types of integrals.

1. Integration by Parts

When we need to find the integral of a product of two functions $f(x)$ and $g(x)$, we can substitute $f(x)$ with $u$ and $g(x)\,dx$ with $dv$. Then, by utilizing the formula for the derivative of a product in differential notation:

This is the Integration by Parts formula. It comes to a rescue when the product of two different functions needs to be integrated and calculating $\displaystyle\int v\,du$ is easier than calculating $\displaystyle\int u\,dv$.

Example 1: Solve $\displaystyle\int x\ln(x)\,dx$

Solution:

Then:

The tricky part in applying integration by parts is how to select $u$ and $dv$ so that finding the value will be easy. When $u\,dv$ is complex, this is especially tricky, as finding the proper $u$ and $dv$ by trial and error takes a lot of time. With practice, one can find proper $u$ and $dv$ easily and faster.

Before applying the rule, keep in mind that:

• We need to be able to compute the integral of $dv$.
• We need to be able to integrate $v\,du$.

If it is known that one of these will fail, we can select other values and avoid wasting time. It is also possible that the integral may be solved more easily by $u$-substitution — testing other integration methods before resorting to integration by parts is usually advised.

Sometimes, one can apply integration by parts several times in succession:

Example 2: Find $\displaystyle\int x^n e^x\,dx$

Solution:

Then:

The power of $x$ became $x^{n-1}$, which means if we continue applying integration by parts, the term $x^n$ will eventually become $x^0$. So we can write:

Note that we continue this until the powers of $x$ and $e$ go to zero.

Example 3 — Reduction Formula: Another cool example is the reduction of $A_n = \displaystyle\int \sin^n x\,dx$:

Therefore:

Now substituting $\cos^2 x$ with $1 - \sin^2 x$:

The index of $A$ becomes smaller each time, so we can lower the index until it becomes $0$ or $1$. If $n$ is odd, we use $A_1 = -\cos x$. Otherwise, $A_0 = x$. We could solve the problem by utilizing half-angle formulas, but when $n$ is large, this becomes cumbersome.

2. Problems

Problem 1. Prove:

(a)

(b)

Problem 2. Integrate

Problem 3. Integrate

3. Solutions

Problem 1(a):

So:

Problem 1(b): Put $n = 5$ in the formula:

Problem 2: Separate $x^2$ and $\dfrac{x}{(x^2+5)^2}\,dx$:

Therefore:

By $u$-substitution on the second term:

Problem 3:

Therefore:

Solving the second term by $u$-substitution: