If `intf(x)dx=psi(x)` , then `intx^5f(x^3")"dx` is equal to

Options

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

Explanation

To solve the integral \( \int x^5 f(x^3) \, dx \), we can use substitution. Let's go through the steps:

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

### Step 2: Express \( x^5 \) in terms of \( t \)
Since \( t = x^3 \), we can express \( x \) in terms of \( t \):
\[ x = t^{1/3} \quad \Rightarrow \quad x^5 = (t^{1/3})^5 = t^{5/3} \]

### Step 3: Substitute \( x^5 \) and \( dx \) into the integral
Now, substituting \( x^5 \) and \( dx \) into the integral:
\[ \int x^5 f(x^3) \, dx = \int t^{5/3} f(t) \cdot \frac{dt}{3x^2} \]
We know \( x^2 = (t^{1/3})^2 = t^{2/3} \), so:
\[ dx = \frac{dt}{3t^{2/3}} \]
Thus, the integral becomes:
\[ \int t^{5/3} f(t) \cdot \frac{dt}{3t^{2/3}} = \frac{1}{3} \int t^{5/3 - 2/3} f(t) \, dt = \frac{1}{3} \int t^{1} f(t) \, dt \]

### Step 4: Simplify the integral
Now, we simplify the integral:
\[ \frac{1}{3} \int t f(t) \, dt \]

### Step 5: Final Result
Thus, we have:
\[ \int x^5 f(x^3) \, dx = \frac{1}{3} \int t f(t) \, dt \]