Integrals Involving Exponential and Logarithmic Functions - Revision history

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Licensing

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Resources

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Logarithmic Differentiation

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Integral of the Inverse function

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Integral of the Exponential function

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2020-08-20T23:41:51Z

Added video link

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[https://youtu.be/D9dqdbCgJQM Integrating Exponential Functions By Substitution - Antiderivatives - Calculus] by The Organic Chemistry Tutor

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 ==Licensing==
 Content obtained and/or adapted from:
-* [https://en.wikibooks.org/wiki/A-level_Mathematics/CIE/Pure_Mathematics_2/Integration Integration, Wikibooks: A-level_Mathematics/CIE/Pure_Mathematics_2/Integration] under a CC BY-SA license
+* [https://en.wikibooks.org/wiki/A-level_Mathematics/CIE/Pure_Mathematics_2/Integration Integration, Wikibooks: A-level Mathematics/CIE/Pure Mathematics 2] under a CC BY-SA license

← Older revision		Revision as of 19:35, 28 October 2021
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	* [https://youtu.be/D9dqdbCgJQM Integrating Exponential Functions By Substitution - Antiderivatives - Calculus] by The Organic Chemistry Tutor		* [https://youtu.be/D9dqdbCgJQM Integrating Exponential Functions By Substitution - Antiderivatives - Calculus] by The Organic Chemistry Tutor
	* [https://en.wikibooks.org/wiki/Calculus/Indefinite_integral Indefinite Integral], WikiBooks: Calculus		* [https://en.wikibooks.org/wiki/Calculus/Indefinite_integral Indefinite Integral], WikiBooks: Calculus
		+
		+	==Licensing==
		+	Content obtained and/or adapted from:
		+	* [https://en.wikibooks.org/wiki/A-level_Mathematics/CIE/Pure_Mathematics_2/Integration Integration, Wikibooks: A-level_Mathematics/CIE/Pure_Mathematics_2/Integration] under a CC BY-SA license

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 |}
-=== Exponential & Logarithm /integrals ===
+== Exponential & Logarithm Integrals ==
 Regarding the derivatives of exponentials and logarithms, we know that:

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 |}
-==Logarithmic Differentiation==
+=== Exponential & Logarithm /integrals ===
-{| WIDTH="100%"
+Regarding the derivatives of exponentials and logarithms, we know that:
-<math>\log(a)+\log(b)=\log(ab)</math>
+* <math>\dfrac{d}{dx} e^x = e^x</math>
-<math>\log\left(\frac{a}{b}\right)=\log(a)-\log(b)</math>
+Reversing these, we get:
-<math>\log(a^n)=n\log(a)</math>
+* <math>\int e^x = e^x + c</math>
-|}
+The natural logarithm takes a modulus input so that it can handle negative numbers.
-See the examples below.
+Note that similar rules apply to any linear expressions that may be composed with these functions:
-====Example 1====
+* <math>\int e^{ax+b} = \frac{1}{a}e^{ax+b} + c</math>
-We shall now prove the validity of the power rule using logarithmic differentiation.
+* <math>\int \frac{1}{ax+b} = \frac{1}{a}\ln|ax+b|</math>
 ==Resources==
 * [https://youtu.be/D9dqdbCgJQM Integrating Exponential Functions By Substitution - Antiderivatives - Calculus] by The Organic Chemistry Tutor
 * [https://en.wikibooks.org/wiki/Calculus/Indefinite_integral Indefinite Integral], WikiBooks: Calculus

@@ Line 16: / Line 16: @@
 ==Resources==
+* [https://youtu.be/D9dqdbCgJQM Integrating Exponential Functions By Substitution - Antiderivatives - Calculus] by The Organic Chemistry Tutor
-==Resources==
+* [https://en.wikibooks.org/wiki/Calculus/Indefinite_integral Indefinite Integral], WikiBooks: Calculus

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 Therefore, since <math>\frac{1}{x}</math> is the derivative of <math>\ln(x)</math> we can conclude that
-{{Calculus/Def|text= <math>\int\frac{dx}{x}=\ln|x|+C</math>}}
+: <math>\int\frac{dx}{x}=\ln|x|+C</math>
 Note that the polynomial integration rule does not apply when the exponent is <math>-1</math> . This technique of integration must be used instead. Since the argument of the natural logarithm function must be positive (on the real line), the absolute value signs are added around its argument to ensure that the argument is positive.

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 we see that <math>e^x</math> is its own antiderivative. This allows us to find the integral of an exponential function:
 : <math>\int e^xdx=e^x+C</math>
 ==Resources==
 [https://youtu.be/D9dqdbCgJQM Integrating Exponential Functions By Substitution - Antiderivatives - Calculus] by The Organic Chemistry Tutor

← Older revision		Revision as of 21:50, 2 October 2021
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		+	===Integral of the Exponential function===
		+	Since
		+	: <math>\frac{d}{dx}e^x=e^x</math>
		+	we see that <math>e^x</math> is its own antiderivative. This allows us to find the integral of an exponential function:
		+	: <math>\int e^xdx=e^x+C</math>
		+
		+	==Resources==
	[https://youtu.be/D9dqdbCgJQM Integrating Exponential Functions By Substitution - Antiderivatives - Calculus] by The Organic Chemistry Tutor		[https://youtu.be/D9dqdbCgJQM Integrating Exponential Functions By Substitution - Antiderivatives - Calculus] by The Organic Chemistry Tutor