Difference between revisions of "Integration Formulas and the Net Change Theorem"
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| − | [https://youtu.be/df1Qr8pepx0 Net Change Theorem, Definite Integral & Rates of Change Word Problems, Calculus] by the Organic Chemistry Tutor | + | ==Indefinite integral identities== |
| + | ===Basic Properties of Indefinite Integrals=== | ||
| + | {{Calculus/Def|text= '''Constant Rule for indefinite integrals'''<br/> | ||
| + | If <math>c</math> is a constant then <math>\int c\cdot f(x)dx=c\int f(x)dx</math>}} | ||
| + | |||
| + | {{Calculus/Def|text= '''Sum/Difference Rule for indefinite integrals'''<br/> | ||
| + | :<math>\int\Big(f(x)+g(x)\Big)dx=\int f(x)dx+\int g(x)dx</math> | ||
| + | :<math>\int\Big(f(x)-g(x)\Big)dx=\int f(x)dx-\int g(x)dx</math>}} | ||
| + | |||
| + | ===Indefinite integrals of Polynomials=== | ||
| + | Say we are given a function of the form, <math>f(x)=x^n</math> , and would like to determine the antiderivative of <math>f</math> . Considering that | ||
| + | :<math>\frac{d}{dx}\frac{1}{n+1}x^{n+1}=x^n</math> | ||
| + | we have the following rule for indefinite integrals: | ||
| + | |||
| + | {{Calculus/Def|text= '''Power rule for indefinite integrals''' | ||
| + | <math>\int x^ndx=\frac{x^{n+1}}{n+1}+C</math> for all <math>n\ne -1</math>}} | ||
| + | |||
| + | ===Integral of the Inverse function=== | ||
| + | To integrate <math>f(x)=\frac{1}{x}</math> , we should first remember | ||
| + | :<math>\frac{d}{dx}\ln(x)=\frac{1}{x}</math> | ||
| + | |||
| + | 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>}} | ||
| + | |||
| + | 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. | ||
| + | |||
| + | ===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: | ||
| + | {{Calculus/Def|text= <math>\int e^xdx=e^x+C</math>}} | ||
| + | |||
| + | ===Integral of Sine and Cosine=== | ||
| + | Recall that | ||
| + | :<math>\frac{d}{dx}\sin(x)=\cos(x)</math> | ||
| + | :<math>\frac{d}{dx}\cos(x)=-\sin(x)</math> | ||
| + | |||
| + | So <math>\sin(x)</math> is an antiderivative of <math>\cos(x)</math> and <math>-\cos(x)</math> is an antiderivative of <math>\sin(x)</math> . Hence we get the following rules for integrating <math>\sin(x)</math> and <math>\cos(x)</math> | ||
| + | |||
| + | {{Calculus/Def|text= <math>\int\cos(x)dx=\sin(x)+C</math></br><math>\int\sin(x)dx=-\cos(x)+C</math>}} | ||
| + | |||
| + | We will find how to integrate more complicated trigonometric functions in the chapter on [[Calculus/Integration techniques|integration techniques]]. | ||
| + | |||
| + | '''Example''' | ||
| + | |||
| + | Suppose we want to integrate the function <math>f(x)=x^4+1+2\sin(x)</math> . An application of the sum rule from above allows us to use the power rule and our rule for integrating <math>\sin(x)</math> as follows, | ||
| + | :{| | ||
| + | |<math>\int f(x)dx</math> | ||
| + | |<math>=\int\Big(x^4+1+2\sin(x)\Big)dx</math> | ||
| + | |- | ||
| + | | | ||
| + | |<math>=\int x^4dx+\int 1\,dx+\int 2\sin(x)dx</math> | ||
| + | |- | ||
| + | | | ||
| + | |<math>=\frac{x^5}{5}+x-2\cos(x)+C</math> . | ||
| + | |} | ||
| + | |||
| + | |||
| + | ==Resources== | ||
| + | * [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 | ||
Revision as of 19:37, 10 October 2021
Contents
Indefinite integral identities
Basic Properties of 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:
Integral of the Inverse function
To integrate , we should first remember
Therefore, since is the derivative of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ln(x)} we can conclude that
Note that the polynomial integration rule does not apply when the exponent is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -1} . 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
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dx}e^x=e^x}
we see that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^x} is its own antiderivative. This allows us to find the integral of an exponential function: Template:Calculus/Def
Integral of Sine and Cosine
Recall that
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dx}\sin(x)=\cos(x)}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dx}\cos(x)=-\sin(x)}
So Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin(x)} is an antiderivative of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos(x)} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\cos(x)} is an antiderivative of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin(x)} . Hence we get the following rules for integrating Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin(x)} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos(x)}
We will find how to integrate more complicated trigonometric functions in the chapter on integration techniques.
Example
Suppose we want to integrate the function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=x^4+1+2\sin(x)} . An application of the sum rule from above allows us to use the power rule and our rule for integrating Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin(x)} as follows,
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int f(x)dx} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\int\Big(x^4+1+2\sin(x)\Big)dx} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\int x^4dx+\int 1\,dx+\int 2\sin(x)dx} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\frac{x^5}{5}+x-2\cos(x)+C} .
Resources
- Indefinite Integral, Wikibooks: Calculus
- Net Change Theorem, Definite Integral & Rates of Change Word Problems, Calculus by the Organic Chemistry Tutor
