Difference between revisions of "Integrals Involving Exponential and Logarithmic Functions"

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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:
 
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>
 
: <math>\int e^xdx=e^x+C</math>
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===Integral of the Inverse function===
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To integrate <math>f(x)=\frac{1}{x}</math> , we should first remember
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:<math>\frac{d}{dx}\ln(x)=\frac{1}{x}</math>
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Therefore, since <math>\frac{1}{x}</math> is the derivative of <math>\ln(x)</math> we can conclude that
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{{Calculus/Def|text= <math>\int\frac{dx}{x}=\ln|x|+C</math>}}
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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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==Resources==
  
 
==Resources==
 
==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

Revision as of 15:57, 2 October 2021

Integral of the Exponential function

Since

we see that is its own antiderivative. This allows us to find the integral of an exponential function:

Integral of the Inverse function

To integrate , we should first remember

Therefore, since is the derivative of we can conclude that

Template:Calculus/Def

Note that the polynomial integration rule does not apply when the exponent is . 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.

Resources

Resources

Integrating Exponential Functions By Substitution - Antiderivatives - Calculus by The Organic Chemistry Tutor