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:

\[e^x,\ e^{2x},\ e^{3x}\ etc\]
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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

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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

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

\[Evaluate:\ \int x^2\ log\ x\ dx\]

Soln:

ILATE

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

\[Evaluate:\ \int x^n\ log\ x\ dx\]

Soln:

ILATE

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