Evaluate:

OR

Evaluate:

Let I =

∵ cos 2x = 2cos² x – 1 and cos 2α = 2cos² α – 1

∴ I =

⇒ I =

As, a² – b² = (a+b)(a-b)

∴ I =

⇒ I =

⇒ I =

⇒ I = 2 sin x + 2 cos α

⇒ I = 2(sin x + x cos α) + C

OR

Let, I =

⇒ I =

⇒ I =

Let, I₁ = and I₂ =

∴ I = I₁ + I₂ …(1)

First we will calculate I₁ and I₂ and then add both to get the answer.

Calculation of I₁ :

Let x² + 2x + 3 = u

⇒ du = 2(x + 1)dx

⇒ (x + 1)dx = du/2

∴ I₁ =

∴ I₁ = + C₁

∴ I₁ = ...(2)

Calculation of I₂ :

I₂ =

⇒ I₂ =

Use the formula:

∴ I₂ =

⇒ I₂ = …(3)

∴ from equation 1, 2 and 3 we have –

I =

∴ I =