Integrate the function: \(\sqrt {1-4x^2}\)
Solution and Explanation
Let \(I=\int \sqrt {1-4x^2}dx=\int \sqrt{(1)^2-(2x)^2}dx\)
Let 2x=t \(\Rightarrow\) 2dx=dt
∴\(I = \frac{1}{2}\sqrt{(1)^2-(t)^2}dt\)
It is known that, \(\int \sqrt{a^2-x^2}dx=\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}\sin^{-1}\frac{x}{a}+C\)
\(\Rightarrow I=\frac{1}{2}\bigg[\frac{t}{2}\sqrt{1-t^2}+\frac{1}{2}\sin^{-1}t\bigg]+C\)
=\(\frac{t}{4}\sqrt{1-t^2}+\frac{1}{4}\sin^{-1}2x+C\)
=\(\frac{2x}{4}\sqrt{1-4x^2}+\frac{1}{4}\sin^{-1}2x+C\)
=\(\frac{x}{2}\sqrt{1-4x^2}+\frac{1}{4}\sin^{-1}2x+C\)
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Concepts Used:
Integrals of Some Particular Functions
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
- ∫1/(x2 + a2) dx = (1/a) tan-1(x/a) + C
- ∫1/√(x2 – a2) dx = log|x + √(x2 – a2)| + C
- ∫1/√(a2 – x2) dx = sin-1(x/a) + C
- ∫1/√(x2 + a2) dx = log|x + √(x2 + a2)| + C
These are tabulated below along with the meaning of each part.
