Maharashtra State BoardHSC Arts 12th Board Exam
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Prove that ∫ 1 a 2 − X 2 D X = 1 2a Log ∣ ∣ ∣ a + X a − X ∣ ∣ ∣ + C - Mathematics and Statistics

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Sum

Prove that `int 1/(a^2 - x^2) dx = 1/"2a" log|(a +x)/(a-x)| + c`

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Solution

`int1/(a^2 - x^2)dx = int1/((a - x)(a + x))dx`

= `1/(2a)int((a - x) + (a + x))/((a - x)(a + x))dx`

= `1/(2a)int(1/(a + x) + 1/(a - x))dx`

= `1/(2a)[int1/(a + x)dx + int1/(a - x)dx]`

= `1/(2a)[log|a + x| + (log|a - x|)/-1] + C = 1/(2a)[log|a + x| - log|a - x|] + C`

= `1/(2a)log |(a + x)/(a - x)| + C`

Concept: Fundamental Theorem of Calculus
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