Recursive version of integration by parts - Revision history

Vipul: /* Examples */

2012-04-08T22:44:30Z

Examples

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	{{#lst:secant-cubed function\|integration by parts}}		{{#lst:secant-cubed function\|integration by parts}}

			===Exponential times cosine function===

			Consider the integration problem:

			<math>\int e^x \cos x \, dx</math>

			We proceed by taking <math>e^x</math> as the part to integrate and <math>\cos x</math> as the part to differentiate. We get:

			<math>\int e^x \cos x \, dx = (\cos x)(e^x) - \int (-\sin x)(e^x) \, dx = e^x \cos x + \int (\sin x)(e^x) \, dx</math>

			To avoid the circular trap, we should pick the exponential function as the part to integrate again, getting:

			<math>\int e^x \cos x \, dx = (\cos x)(e^x) + (\sin x)(e^x) - \int (\cos x)(e^x) \, dx</math>

			Now, we take <math>I</math> as the antiderivative where the above holds without additive constants, and get:

			<math>I = e^x(\cos x + \sin x) - I</math>

			Rearranging, we get:

			<math>2I = e^x(\cos x + \sin x)</math>

			This gives us:

			<math>I = \frac{e^x(\cos x + \sin x)}{2}</math>

			So the general integral is:

			<math>\int e^x \cos x \, dx = \frac{e^x(\cos x + \sin x)}{2} + C</math>

Vipul at 22:35, 8 April 2012

2012-04-08T22:35:50Z

← Older revision		Revision as of 22:35, 8 April 2012
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	# Solve the linear equation in <math>I</math>.		# Solve the linear equation in <math>I</math>.
	# The general indefinite integral is this solution plus an arbitrary additive constant.		# The general indefinite integral is this solution plus an arbitrary additive constant.

			<center>{{#widget:YouTube\|id=A91mhGOxYsA}}</center>

	==Examples==		==Examples==

Vipul at 16:57, 8 April 2012

2012-04-08T16:57:19Z

← Older revision		Revision as of 16:57, 8 April 2012
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	==General description of technique==		==General description of technique==

	~~{{fillin}}~~		The general procedure is as follows:

			# Start doing the [[integration by parts]].
			# If necessary, apply integration by parts twice OR use some trigonometric identity with the ultimate goal of seeing the original integration appear ''again''. Note that you should take care to avoid the [[Integration by parts#Repeated use of integration by parts and the circular trap\|circular trap]].
			# Denote by <math>I</math> the particular choice of antiderivative for which the left and right sides are equal on the nose,i.e., not just up to additive constants.
			# Solve the linear equation in <math>I</math>.
			# The general indefinite integral is this solution plus an arbitrary additive constant.

	==Examples==		==Examples==

Vipul: Created page with "==General description of technique== {{fillin}} ==Examples== ===Sine-squared function=== {{further|Sine-squared function#Integration}} There are many ways of integrating..."

2011-09-19T21:16:51Z

Created page with "==General description of technique== {{fillin}} ==Examples== ===Sine-squared function=== {{further|Sine-squared function#Integration}} There are many ways of integrating..."

New page

==General description of technique==

{{fillin}}

==Examples==

===Sine-squared function===

{{further|[[Sine-squared function#Integration]]}}

There are many ways of integrating <math>\sin^2</math>. One of these uses the recursive version of integration by parts. This method is given below:

{{#lst:sine-squared function|integration by parts}}

===Secant-cubed function===

{{further|[[Secant-cubed function#Integration]]}}

{{#lst:secant-cubed function|integration by parts}}