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Question

Mathematics Question on integral

If ∫f(x)dx=Ψ(x)\int f\left(x\right) dx =\Psi\left(x\right) then ∫x5f(x3)dx\int x^{5} f\left(x^{3}\right)dx is equal to

A

13x3Ψ(x3)−3∫x3Ψ(x3)dx+C\frac{1}{3}x^{3}\Psi \left(x^{3}\right)-3 \int x^{3}\Psi \left(x^{3}\right)dx +C

B

13x3Ψ(x3)−∫x3Ψ(x3)dx+C\frac{1}{3}x^{3}\Psi \left(x^{3}\right)- \int x^{3}\Psi \left(x^{3}\right)dx +C

C

13[x3Ψ(x3)−3∫x3Ψ(x3)dx]+C\frac{1}{3} [ x^{3}\Psi \left(x^{3}\right)-3 \int x^{3}\Psi \left(x^{3}\right)dx] +C

D

13[x3Ψ(x3)−∫x2Ψ(x3)dx]+C\frac{1}{3} [ x^{3}\Psi \left(x^{3}\right)- \int x^{2}\Psi \left(x^{3}\right)dx] +C

Answer

13x3Ψ(x3)−∫x3Ψ(x3)dx+C\frac{1}{3}x^{3}\Psi \left(x^{3}\right)- \int x^{3}\Psi \left(x^{3}\right)dx +C

Explanation

Solution

Let x3=ux^3 = u, then 3x2dx=du3x^2 dx = du
Also suppose ∫f(x)dxΨ(x)\int f(x) dx \Psi (x)
Now ∫x5f(x3)dx=13∫uf(u)du\int x^{5} f\left(x^{3}\right)dx = \frac{1}{3}\int u f\left(u\right)du
=13[u∫f(u)dx−∫(∫f(u))du]= \frac{1}{3}\left[u \int f\left(u\right)dx-\int\left(\int f\left(u\right)\right)du \right]
=13[x3Ψ(x3)−∫x2Ψ(x3)dx]+C= \frac{1}{3} [ x^{3}\Psi \left(x^{3}\right)- \int x^{2}\Psi \left(x^{3}\right)dx] +C