Question

If `int(2e^(5x) + e^(4x) - 4e^(3x) + 4e^(2x) + 2e^x)/((e^(2x) + 4)(e^(2x) - 1)^2)dx = tan^-1(e^x/a) - 1/(b(e^(2x) - 1)) + C`, where C is constant of integration, then value of a + b is equal to ______.

Options

• 2.00

• 3.00

• 4.00

• 5.00

Answer 1

If `int(2e^(5x) + e^(4x) - 4e^(3x) + 4e^(2x) + 2e^x)/((e^(2x) + 4)(e^(2x) - 1)^2)dx = tan^-1(e^x/a) - 1/(b(e^(2x) - 1)) + C`, where C is constant of integration, then value of a + b is equal to 4.00.

Explanation:

I = `int{(2(e^x)^4 + (e^x)^3 - 4(e^x)^2 + 4e^x + 2)/(((e^x)^2 + 4)((e^x)^2 - 1)^2)}d(e^x)`

Put e^x = t

I = `int ((2t^4 - 4t^2 + 2) + (t^3 + 4t))/((t^2 + 4)(t^2 - 1)^2)dt`

= `int 2/(t^2 + 4) + t/(t^2 - 1)^2 dt`

= `tan^-1(t/2) - 1/(2(t^2 - 1)) + c`

∴ a + b = 4