Maharashtra State BoardHSC Arts 12th Board Exam
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Prove that : ∫√(a2−x2)dx=x/2 √(a^2−x^2)=a^2/2 sin^−1(x/a)+c - Mathematics and Statistics

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Prove that : `int sqrt(a^2-x^2)dx=x/2sqrt(a^2-x^2)=a^2/2sin^-1(x/a)+c`

 

 

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Solution

`intsqrt(a^2-x^2)dx`

Substitute x = asinθ ...(i)

dx = acosθ dθ
The integral becomes

`intsqrt(a^2-a^2sin^2theta)acostheta d theta`

`=intasqrt(1-sin^2theta)acostheta d theta`

`=a^2intcos^2theta d theta`

`=a^2int(1+cos2theta)/2 d theta`

`=a^2[int1/2d theta+int(cos2theta)/2d theta]`

`=(a^2theta)/2+a^2/4sin2theta+C`

`"From "(i) ,theta=sin^-1(x/a),sin2theta=2sinthetacostheta=2(x/a)sqrt(1-(x^2/a^2))=(2x)/a^2sqrt(a^2-x^2)`

Substituting these values, we get

 

`intsqrt(a^2-x^2)dx=a^2/2sin^-1(x/a)+a^2/4xx(2x)/a^2sqrt(a^2-x^2)+C`

`=a^2/2sin^-1(x/a)+x/2sqrt(a^2-x^2)+C " (Proved)"`

 

Concept: Evaluation of Definite Integrals by Substitution
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