Find :

We have to find the integral of but first we will convert this trigonometric function into algebraic function by substitution which is,

Put e^x = t, so,

= e^x,

Hence e^x dx = dt, putting in the function we get,

I =

We will solve this function by using partial fractions.

A(t + 2)(t - 1) + B(t + 2) + C(t - 1)² = t⁰1 + t¹0 + t²0

At² + At - 2A + Bt + 2B + Ct² + C – Ct = t⁰1 + t¹0 + t²0

A + C = 0, A + B – C = 0, –2A + 2B + C = 1

B = 2C, 2C + 4C + C = 1,

C = , A = , B =

Now making a function into partial fractions, we get,

I =

I = ln – ln + C