Question

Evaluate the following integrals : `int sqrt((e^(3x) - e^(2x))/(e^x + 1)).dx`

Answer 1

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 e^x = t

∴ e^x 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.