Difference between revisions of "Integration Formulas and the Net Change Theorem"

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<math>\int\cos(x)dx=\sin(x)+C</math></br><math>\int\sin(x)dx=-\cos(x)+C</math>
 
<math>\int\cos(x)dx=\sin(x)+C</math></br><math>\int\sin(x)dx=-\cos(x)+C</math>
 
</blockquote>
 
</blockquote>
 
We will find how to integrate more complicated trigonometric functions in the chapter on [[Calculus/Integration techniques|integration techniques]].
 
  
 
'''Example'''
 
'''Example'''
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|<math>=\frac{x^5}{5}+x-2\cos(x)+C</math> .
 
|<math>=\frac{x^5}{5}+x-2\cos(x)+C</math> .
 
|}
 
|}
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 +
==Recognizing Derivatives and Reversing Derivative Rules==
 +
If we recognize a function <math>g(x)</math> as being the derivative of a function <math>f(x)</math> , then we can easily express the antiderivative of <math>g(x)</math> :
 +
 +
<math>\int g(x)dx=f(x)+C</math>
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 +
For example, since
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 +
<math>\frac{d}{dx}\sin(x)=\cos(x)</math>
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 +
we can conclude that
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 +
<math>\int\cos(x)dx=\sin(x)+C</math>
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 +
Similarly, since we know <math>e^x</math> is its own derivative,
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<math>\int e^xdx=e^x+C</math>
 +
 +
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The power rule for derivatives can be reversed to give us a way to handle integrals of powers of <math>x</math> . Since
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<math>\frac{d}{dx}x^n=nx^{n-1}</math>
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 +
we can conclude that
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<math>\int nx^{n-1}dx=x^n+C</math>
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 +
or, a little more usefully,
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<math>\int x^ndx=\frac{x^{n+1}}{n+1}+C</math>
  
 
==Resources==
 
==Resources==
 
* [https://en.wikibooks.org/wiki/Calculus/Indefinite_integral Indefinite Integral], Wikibooks: Calculus
 
* [https://en.wikibooks.org/wiki/Calculus/Indefinite_integral Indefinite Integral], Wikibooks: Calculus
 
* [https://youtu.be/df1Qr8pepx0 Net Change Theorem, Definite Integral & Rates of Change Word Problems, Calculus] by the Organic Chemistry Tutor
 
* [https://youtu.be/df1Qr8pepx0 Net Change Theorem, Definite Integral & Rates of Change Word Problems, Calculus] by the Organic Chemistry Tutor

Revision as of 19:44, 10 October 2021

Indefinite integral identities

Basic Properties of Indefinite Integrals

Constant Rule for indefinite integrals

If is a constant then }}


Sum/Difference Rule for indefinite integrals

}}

Indefinite integrals of Polynomials

Say we are given a function of the form, , and would like to determine the antiderivative of . Considering that

we have the following rule for indefinite integrals:

Power rule for indefinite integrals

for all

Integral of the Inverse function

To integrate , we should first remember

Therefore, since is the derivative of we can conclude that

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.

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 Sine and Cosine

Recall that

So is an antiderivative of and is an antiderivative of . Hence we get the following rules for integrating and


Example

Suppose we want to integrate the function . An application of the sum rule from above allows us to use the power rule and our rule for integrating as follows,

.

Recognizing Derivatives and Reversing Derivative Rules

If we recognize a function as being the derivative of a function , then we can easily express the antiderivative of  :

For example, since

we can conclude that

Similarly, since we know is its own derivative,


The power rule for derivatives can be reversed to give us a way to handle integrals of powers of . Since

we can conclude that

or, a little more usefully,

Resources