Evaluate the following integral:

Let us assume

.....….equation 1

By property we know that

Thus

.....equation 2

Adding equation 1 and equation 2

We know

……equation

Let cosx = y

Differentiating both sides

– sinxdx = dy

sinxdx = – dy

for x = 0

cos0 = y

1 = y

For x = π

cosπ = y

– 1 = y

Substituting equation 3 becomes

2I = π[{3(1) – (1)³} – {3( – 1) – ( – 1)³}]/3

2I = π[2 – { – 3 + 1}]/3

2I = π[2 + 2]/3

I = 2π/3