Integrate the function- -1-4 x2 d x

Let I =∫√(1 4x^2)dx =∫√(4 ( 1/4 x^2 ))dx =2 ∫√( ( ( 1/2 )^2 x^2 ))dx [ ∵∫√(a^2 x^2)dx =x/2√(a^2 x^2)+a^2/2sin^ 1x/a+C ] =2.x/2√(1/4 x^2)+1/4.2/2sin^ 1x ...

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Standard XII

Mathematics

Integration by Parts

Integrate the...

Question

Integrate the function.
$\int \sqrt{1 - 4 x^{2}} d x .$

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Solution

Let $I = \int \sqrt{1 - 4 x^{2}} d x$
$= \int \sqrt{4 (\frac{1}{4} - x^{2})} d x = 2 \int \sqrt{({(\frac{1}{2})}^{2} - x^{2})} d x [∵ \int \sqrt{a^{2} - x^{2}} d x = \frac{x}{2} \sqrt{a^{2} - x^{2}} + \frac{a^{2}}{2} s i n^{- 1} \frac{x}{a} + C] = 2. \frac{x}{2} \sqrt{\frac{1}{4} - x^{2}} + \frac{1}{4} . \frac{2}{2} s i n^{- 1} \frac{x}{\frac{1}{2}} + C \Rightarrow I = \frac{x}{2} \sqrt{1 - 4 x^{2}} + \frac{1}{4} s i n^{- 1} (2 x) + C$

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Integrate the function. ∫√1−4x2dx.

Let I=∫√1−4x2dx =∫√4(14−x2)dx=2∫√((12)2−x2)dx[∵∫√a2−x2dx=x2√a2−x2+a22sin−1xa+C]=2.x2√14−x2+14.22sin−1x12+C⇒I=x2√1−4x2+14sin−1(2x)+C

Integrate the function.
$\int \sqrt{1 - 4 x^{2}} d x .$