Evaluate: <span l

dx

∵ - x = x( - 1) = x (x – 1) (x + 1)

- x > 0 for -1 < x < 0

- x < 0 for 0 < x < 1

- x > 0 for 1 < x < 2

NOTE: Sign is only being considered for the range -1≤ x ≤ 2

dx + dx + dx

dx – dx + dx

I = - +

I =

I =

The required value is .

OR

Let, I = …(1)

Using the property of definite Integral: =

I =

I = …(2)

_{Adding equation 1 and 2 we get –}

2I =

2I =

Putting Cos x = t

-Sin x dx = dt

When x = 0 ⇒ t = 1

x = ⇒ t = -1

⇒ 2I = {∵ }

⇒ 2I = π

∴ 2I = π{tan^-1(1) – tan^-1(-1)} {∵ tan^-1(1) = π/4}

⇒ 2I = π(π/4 – (-π/4))

⇒ I =

Hence,