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Question
Evaluate the following : `int sqrt((10 + x)/(10 - x)).dx`
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Solution
Let I = `int sqrt((10 + x)/(10 - x)).dx`
= `int sqrt((10 + x)/(10 - x) xx (10 + x)/(10 + x)).dx`
= `int (10 + x)/sqrt(100 - x^2).dx`
= `int (10)/sqrt(100 - x^2).dx + int x/sqrt(100 - x^2).dx`
= `10 int (1)/sqrt(10^2 - x^2).dx + (1)/(2) int (2x)/sqrt(100 - x^2).dx`
= I1 + I2 ...(Let)
I1 = `10 int (1)/sqrt(10^2 - x^2).dx`
= `10 sin^-1 (x/10) + c_1`
In I2, put 100 – 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(100 - x^2) + c_2`
I = `10 sin^-1 (x/10) - sqrt(100 - x^2) + c`.
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