Question:
Integrate the function: \(\sqrt{1-4x-x^2}\)
Integrate the function: \(\sqrt{1-4x-x^2}\)
Updated On: Oct 4, 2023
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Solution and Explanation
Let \(I=\int \sqrt{1-4x-x^2}dx\)
=\(\int \sqrt{1-(x^2+4x+4-4)}dx\)
=\(\int \sqrt{1+4-(x+2)^2}dx\)
=\(\int\sqrt{(\sqrt5)^2-(x+2)^2}dx\)
It is known that,\(\int \sqrt{a^2-x^2}dx=\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}\sin^{-1}\frac{x}{a}+C\)
∴\(I=\frac{(x+2)}{2}\sqrt{1-4x-x^2}+\frac{5}{2}\sin^{-1}\bigg(\frac{x+2}{\sqrt 5}\bigg)+C\)
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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
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These are tabulated below along with the meaning of each part.
