5.1 INTEGRATION BY PARTS

$Integrals\ of\ the\ form\ \int x\ sin\ nx\ dx,\ \int x\ cos\ nx\ dx,\ \int x\ e^{nx}\ dx,$
$\int x^n\ log\ x\ dx\ and\ \int log\ x\ dx\ Simple\ Problems$

Introduction:

When the integrand is a product of two functions and the method of decomposition or substitution can not be applied, then the method of by parts is used.

Integraiton by parts formula:

$\int u\ dv = uv – \int v\ du$

The above formula is used by taking proper choice of ‘u’ and ‘dv’. ‘u’ should be chosen based on thefollowing order of Preference. Simply remember ILATE

1. Inverse trigonometric functions:

${sin}^{-1}x,\ {cos}^{-1}x,\ {tan}^{-1}x,\ etc$

2. Logarithmic functions: log x

3. Algebraic functions:

$1,\ x\ x^2,\ x^3\ etc$

4. Trigonometric functions: sin x, cos x, tan x, etc.

5. Exponential functions:

$e^x,\ e^{2x},\ e^{3x}\ etc$

Example:

$Evaluate:\ \int x\ sin\ x\ dx$

Soln:

ILATE

u = x                                     dv  = sin x dx

$\frac{du}{dx} = \frac{d}{dx} (x)\ \hspace{2cm}\ \int dv = \int sinx\ dx$
$\frac{du}{dx} = 1\ \hspace{2cm}\ v =- cos x$
$\ du = dx\ \hspace{5cm}$
$\int u\ dv = uv – \int v\ du$
$\int x\ sin\ x\ dx = x ( – cos x) – \int – cos x\ dx$

= -x cos x  +  sin x + c

Example:

$Evaluate:\ \int x\ e^x\ dx$

Soln:

ILATE

$u= x\ \hspace{2cm}\ dv = e^x\ dx$
$\frac{du}{dx} = \frac{d}{dx} (x)\ \hspace{2cm}\ \int dv = \int e^x\ dx$
$\frac{du}{dx} = 1\ \hspace{2cm}\ v = e^x$
$\ du = dx\ \hspace{5cm}$
$\int u\ dv = uv – \int v\ du$
$\int x\ e^x dx = x\ e^x – \int e^x\ dx$
$= x\ e^x – e^x +c$

Example:

$Evaluate:\ \int x\ e^{nx}\ dx$

Soln:

$ILATE$
$u= x\ \hspace{2cm}\ dv = e^{nx}\ dx$
$\frac{du}{dx} = \frac{d}{dx} (x)\ \hspace{2cm}\ \int dv = \int e^{nx}\ dx$
$\frac{du}{dx} = 1\ \hspace{2cm}\ v = \frac{e^{nx}}{n}$
$\ du = dx\ \hspace{5cm}$
$\int u\ dv = uv – \int v\ du$
$\int x\ e^{nx} dx = x\ \frac{e^{nx}}{n} – \int \frac{e^{nx}}{n}\ dx$
$= x\ \frac{e^{nx}}{n} – \frac{e^{nx}}{n^2}\ dx\ + c$

Example:

$Evaluate:\ \int log\ x\ dx$

Soln:

$\int log\ x\ dx = \int 1.\ log\ x\ dx$

ILATE

$u= log\ x\ \hspace{2cm}\ dv = 1\ dx$
$\frac{du}{dx} = \frac{d}{dx} ( log\ x)\ \hspace{2cm}\ \int dv = \int 1\ dx$
$\frac{du}{dx} = \frac{1}{x}\ \hspace{2cm}\ v = x$
$\ du = \frac{1}{x}\ dx\ \hspace{5cm}$
$\int u\ dv = uv – \int v\ du$
$\int log\ x\ dx = x\ log\ x- \int x\ (\frac{1}{x})\ dx$
$= x\ log\ x- \int 1\ dx$
$= x\ log\ x- x + c$

Example:

$Evaluate:\ \int x\ sin\ 2x\ dx$

Soln:

ILATE

$u= x\ \hspace{2cm}\ dv =sin\ 2x\ dx$
$\frac{du}{dx} = \frac{d}{dx} (x)\ \hspace{2cm}\ \int dv = \int sin\ 2x\ dx$
$\frac{du}{dx} = 1\ \hspace{2cm}\ v = -\frac{cos\ 2x}{2}$
$\ du = dx\ \hspace{5cm}$
$\int u\ dv = uv – \int v\ du$
$\int x\ sin\ 2x\ dx = – x\ \frac{cos\ 2x}{2} + \int \frac{cos\ 2x}{2}\ dx$
$= – x\ \frac{cos\ 2x}{2} + \frac{1}{2}(\frac{sin\ 2x}{2})\ + c$
$= – x\ \frac{cos\ 2x}{2} +\frac{sin\ 2x}{4}\ + c$

Example:

$Evaluate:\ \int x\ cos\ 5x\ dx$

Soln:

ILATE

$u= x\ \hspace{2cm}\ dv =cos\ 5x\ dx$
$\frac{du}{dx} = \frac{d}{dx} (x)\ \hspace{2cm}\ \int dv = \int cos\ 5x\ dx$
$\frac{du}{dx} = 1\ \hspace{2cm}\ v = \frac{sin\ 5x}{5}$
$\ du = dx\ \hspace{5cm}$
$\int u\ dv = uv – \int v\ du$
$\int x\ cos\ 5x\ dx = x\ \frac{sin\ 5x}{5} – \int \frac{sin\ 5x}{5}\ dx$
$= x\ \frac{sin\ 5x}{5} – \frac{1}{5}(\frac{cos\ 5x}{5})\ + c$
$= x\ \frac{sin\ 5x}{5} +\frac{cos\ 5x}{25}\ + c$

Example:

$Evaluate:\ \int x^2\ log\ x\ dx$

Soln:

ILATE

$u= log\ x\ \hspace{2cm}\ dv = x^2\ dx$
$\frac{du}{dx} = \frac{d}{dx} ( log\ x)\ \hspace{2cm}\ \int dv = \int x^2\ dx$
$\frac{du}{dx} = \frac{1}{x}\ \hspace{2cm}\ v = \frac{x^3}{3}$
$\ du = \frac{1}{x}\ dx\ \hspace{5cm}$
$\int u\ dv = uv – \int v\ du$
$\int x^2\ log\ x\ dx = log\ x\ \frac{x^3}{3} – \int \frac{x^3}{3}\ (\frac{1}{x})\ dx$
$= log\ x\ \frac{x^3}{3} – \frac{1}{3}\ \int x^2\ dx$
$= log\ x\ \frac{x^3}{3} – \frac{1}{3}\ \frac{x^3}{3}\ + c$
$= log\ x\ \frac{x^3}{3} – \frac{x^3}{9}\ + c$

Example:

$Evaluate:\ \int x^n\ log\ x\ dx$

Soln:

ILATE

$u= log\ x\ \hspace{2cm}\ dv = x^n\ dx$
$\frac{du}{dx} = \frac{d}{dx} ( log\ x)\ \hspace{2cm}\ \int dv = \int x^n\ dx$
$\frac{du}{dx} = \frac{1}{x}\ \hspace{2cm}\ v = \frac{x^{n+1}}{n+1}$
$\ du = \frac{1}{x}\ dx\ \hspace{5cm}$
$\int u\ dv = uv – \int v\ du$
$\int x^n\ log\ x\ dx = log\ x\ \frac{x^{n+1}}{n+1} – \int \frac{x^{n+1}}{n+1}\ (\frac{1}{x})\ dx$
$= log\ x\ \frac{x^{n+1}}{n+1} – \frac{1}{n+1}\ \int x^{n}\ dx$
$= log\ x\ \frac{x^{n+1}}{n+1} – \frac{1}{(n+1)}\ \frac{x^{n+1}}{(n+1)}\ + c$
$= log\ x\ \frac{x^{n+1}}{n+1} – \frac{1}{(n+1)^2}\ x^{n+1}\ + c$