If ∫ f ( x ) d x = ψ ( x ) , then ∫ x 5 f ( x 3 ) d x , is equal to

If ∫f(x)dx=ψ(x), then ∫x5f(x3)dx, is equal to 
  1. 13x3ψ(x3)−∫x2ψ(x3)dx+c
  2. 13[x3ψ(x3)−∫x3ψ(x3)dx]+c
  3. 13[x3ψ(x3)−∫x2ψ(x3)dx]+c
  4. 13x3ψ(x3)−3∫x3ψ(x3)dx+c

Solution

∫x5fx3dx =∫x3.x2fx3dx  

Applying integration by parts, we get

=x3∫x2fx3dx-∫ddxx3∫x2fx3dx

=13x3ψx3-∫x2ψx3dx+c, where c is the constant of integration.

Asked in: JEE Main 2013 (07 Apr)

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