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Question
Evaluate the following integrals : `int sqrt((e^(3x) - e^(2x))/(e^x + 1)).dx`
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Solution
Let I = `int sqrt((e^(3x) - e^(2x))/(e^x + 1)).dx`
= `int sqrt((e^(2x)(e^x - 1))/(e^x + 1)).dx`
= `int e^xsqrt((e^x - 1)/(e^x + 1)).dx`
Put ex = t
∴ ex dx = dt
∴ I = `int sqrt((t - 1)/(t + 1))dt`
= `int sqrt((t - 1)/(t + 1) xx (t - 1)/(t - 1))dt`
= `int sqrt(((t - 1)^2)/(t^2 - 1)dt`
= `int (t - 1)/sqrt(t^2 - 1)dt`
= `(1)/(2) int (2t)/sqrt(t^2 - 1)dt - int (1)/sqrt(t^2 - 1)dt`
= I1 – I2
In I1, put t2 – 1 = θ
∴ 2t dt = dθ
∴ I1 = `(1)/(2)int (dθ)/sqrt(θ)`
= `(1)/(2) int θ^(-1/2) dθ`
= `(1)/(2) (θ^(1/2))/((1/2)) + c_1`
= `sqrt(θ) + c_1`
= `sqrt(t^2 - 1) + c_1`
= `sqrt(e^(2x) - 1) + c_1`
and I2 = `int (1)/sqrt(t^2 - 1)dt`
= `log|t + sqrt(t^2 - 1)| + c_2`
= `log|e^x + sqrt(e^(2x) - 1)| + c_2`
∴ I = `sqrt(e^(2x) - 1) - log|e^x + sqrt(e^(2x) - 1) + c`, where c = c1 + c2.
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