Question:
Integrate the function: \(\sqrt{4-x^2}\)
Integrate the function: \(\sqrt{4-x^2}\)
Updated On: Oct 4, 2023
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Solution and Explanation
Let\(I = \int \sqrt{4-x^2}dx = \int \sqrt{(2)^2-(x)^2}dx\)
It is known that,\(\int \sqrt {a^2-x^2}dx = \frac{x}{2} \sqrt {a^2-x^2} \frac{a^2}{2}\sin^{-1}\frac{x}{x}+C\)
∴\(I = \frac{x}{2}\sqrt{4-x^2}+\frac{4}{2}\sin^{-1}\ \frac{x}{2}+C\)
=\(\frac{x}{2}\sqrt{4-x^2}+2\sin^{-1}\frac{x}{2}+C\)
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There are many important integration formulas which are applied to integrate many other standard integrals. In this article, we will take a look at the integrals of these particular functions and see how they are used in several other standard integrals.
Integrals of Some Particular Functions:
- ∫1/(x2 – a2) dx = (1/2a) log|(x – a)/(x + a)| + C
- ∫1/(a2 – x2) dx = (1/2a) log|(a + x)/(a – x)| + C
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These are tabulated below along with the meaning of each part.
