Caluclus: Table of contents

Integration by Parts

Introduction

Finding the indefinite integral of the function \((x\cos x+\sin x)\) is difficult on the first sight. However, one hint changes the world:

You may now try to solve the problem yourself, and then look at the following example that solves the problem.

Example 1 Find \(\int \left(x\cos x+\sin x\right)\,dx\).

The following problems are similar in nature, and it is a good idea to work on them before proceeding to the theory of integration by parts.

Example 2 Find \(\int \left(x^2\cos x+2x\sin x\right)\,dx\).

Example 3 Find \(\int \left(x^2+2x\right)e^x\,dx\).

We may now solve a problem similar to the one from Example 1:

Example 4 Find \(\int x\cos x\,dx\).

The method of integration by parts

Theorem (Integration by parts)

If \(u\) and \(v\) are differentiable functions then

\[\int u(x)\cdot v^{\prime}(x)\,dx=u(x)\cdot v(x)-\int v(x)\cdot u^{\prime}(x)\,dx.\]

Let us color the integral on the left in blue, and the integral on the right in green.

\(\int u(x)\cdot v^{\prime}(x)\,dx\)\(= u(x)\cdot v(x)-\) \(\int v(x)\cdot u^{\prime}(x)\,dx\).

The idea of the integration by parts is to start with difficult blue integral, and end up with an easy green integral. In Example 4, the integral \(\int x\cos x\,dx\) was blue, and \(\int \sin x\,dx\) was green.

Example 5 Find \(\int x e^x\,dx\).

Example 6 Find \(\int x \sin x\,dx\).

Example 7 Find \(\int \ln x\,dx\).

Example 8 Find \(\int \sin x\cdot e^x\,dx\).

Practice problems

Problem 1. Determine the indefinite integral \[\int x\arctan x\,dx.\]

Problem 2. Determine the integral \[\int_0^{\frac{\pi}2} x\sin x\,dx.\]

Problem 3. Determine the indefinite integral \[\int_1^5 x\ln x\,dx.\]

Problem 4. Find the integral \[\int_{-5}^5 x^4\cdot \sin\left(x^3\right)\,dx.\]

Problem 5. Let \(k\) be the circle of radius \(6\) with center at the origin. Let \(A(3,4)\), \(B(3,0)\), and \(C(5,0)\) be three points such that \(A\) and \(C\) are on the circle and \(B\) is the projection of \(A\) to the \(x\) axis. Find the area of the figure bounded by the segments \(AB\) and \(BC\), and the arc \(\stackrel \frown{AC}\) of the circle \(k\).