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