If `intf(x)= Psi(x)dx+C` then
`intx^(5)f(x^3)dx` is equal to

Options

A. `1/3[x^(3)Psi(x^(3))-intx^(2)Psi(x^(3))dx]+C`
B. `1/3x^(2)Psi(x^(3)-3intx^(3)dx+C)`
C. `1/3x^(3)Psi(x^(3))-intx^(2)Psi(x^(2))dx+C`
D. `1/3[x^(3)Psi(x^(3))-intx^(3)Psi(x^(3))dx]+C`

Explanation

To solve the integral \( \int x^5 f(x^3) \, dx \), we will use substitution and integration by parts. Here’s a step-by-step solution:

### Step 1: Substitution
Let \( t = x^3 \). Then, we differentiate both sides:
\[ \frac{dt}{dx} = 3x^2 \quad \Rightarrow \quad dt = 3x^2 \, dx \quad \Rightarrow \quad dx = \frac{dt}{3x^2} \]

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