Math 22 Antiderivatives and Indefinite Integrals
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Antiderivatives
A function is an antiderivative of a function when for every in the domain of , it follows that
The antidifferentiation process is also called integration and is denoted by (integral sign). is the indefinite integral of
If for all , we can write: for is a constant.
Basic Integration Rules
for is a constant.
for
Exercises 1 Find the indefinite integral
1)
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2)
| Solution: |
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3)
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4)
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| Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \int 5x^{-3}dx=5\int x^{-3}dx=5{\frac {x^{-3+1}}{-3+1}}+C={\frac {-5}{2}}x^{-2}+C} |
Exercises 2 Solve the initial value problems, given:
5) and
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| Notice |
| So, |
| we are given , so |
| Hence, |
| Therefore, |
6) and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(-1)=-6}
| Solution: |
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| Notice |
| So, |
| we are given Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(-1)=-6} , so Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (-1)^{3}+4(-1)+C=-6} |
| Hence, |
| Therefore, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(x)=x^{3}+4x-1} |
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