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Question
Evaluate the following : `int sqrt((2 + x)/(2 - x)).dx`
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Solution
Let I = `int sqrt((2 + x)/(2 - x)).dx`
= `int sqrt((2 + x)/(2 - x) xx (2 + x)/(2 + x)).dx`
= `int (2 + x)/sqrt(4 - x^2).dx`
= `int (2)/sqrt(4 - x^2).dx + int x/sqrt(4 - x^2).dx`
= `2 int (1)/sqrt(2^2 - x^2).dx + (1)/(2) int (2x)/sqrt(4 - x^2).dx`
= I1 + I2 ...(Let)
I1 = `2 int (1)/sqrt(2^2 - x^2).dx`
= `2 sin^-1 (x/2) + c_1`
In I2, put 4 – x2 = t
∴ – 2x dx = dt
∴ 2x dx = – dt
I2 = `-(1)/(2) int t^(-1/2) dt`
= `-(1)/(2).t^(1/2)/((1/2)) + c_2`
= `- sqrt(4 - x^2) + c_2`
I = `2 sin^-1 (x/2) - sqrt(4 - x^2) + c`.
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