If `intf(x)dx=phi(x)i.e.phi` is a function such that `phi'(x) = f(x)`, then `int x^(9) f(x^(5)) dx` is equal to

Options

A. `(1)/(5)[x^(5)phi(x^(5))-intx^(4)phi(x^(5))dx]+C`
B. `(1)/(5)x^(5)phi(x^(5))-5intx^(5)phi(x^(5))dx+C`
C. `(1)/(5)x^(5)phi(x^(5))-intx^(4)phi(x^(5))dx+C`
D. `(1)/(5) [x^(5)phi(x^(5))-intx^(5)phi(x^(5))dx]+C`

Answer

To solve the integral \( \int x^9 f(x^5) \, dx \), we can follow these steps:

### Step 1: Substitution
Let \( t = x^5 \). Then, we differentiate to find \( dt \):
\[ dt = 5x^4 \, dx \quad \Rightarrow \quad dx = \frac{dt}{5x^4} \]

### Step 2: Express \( x^9 \) in terms of \( t \)
Since \( t = x^5 \), we can express \( x \) in terms of \( t \):
\[ x = t^{1/5} \quad \Rightarrow \quad x^9 = (t^{1/5})^9 = t^{9/5} \]

### Step 3: Substitute into the integral
Now we can substitute \( x^9 \) and \( dx \) into the integral:
\[ \int x^9 f(x^5) \, dx = \int t^{9/5} f(t) \cdot \frac{dt}{5x^4} \]
Since \( x^4 = (t^{1/5})^4 = t^{4/5} \), we have:
\[ dx = \frac{dt}{5t^{4/5}} \]
Thus, the integral becomes:
\[ \int t^{9/5} f(t) \cdot \frac{dt}{5t^{4/5}} = \frac{1}{5} \int t^{9/5 - 4/5} f(t) \, dt = \frac{1}{5} \int t^{1} f(t) \, dt \]

### Step 4: Integration by Parts
Now we can apply integration by parts where we let:
- \( u = t \) and \( dv = f(t) \, dt \)
- Then, \( du = dt \) and \( v = \phi(t) \) (since \( \int f(t) \, dt = \phi(t) \))

Using integration by parts:
\[ \int u \, dv = uv - \int v \, du \]
We have:
\[ \frac{1}{5} \left( t \phi(t) - \int \phi(t) \, dt \right) \]

### Step 5: Substitute back \( t = x^5 \)
Now we substitute back \( t = x^5 \):
\[ = \frac{1}{5} \left( x^5 \phi(x^5) - \int \phi(x^5) \, dt \right) \]

### Final Answer
Thus, the final expression for the integral \( \int x^9 f(x^5) \, dx \) is:
\[ \frac{1}{5} \left( x^5 \phi(x^5) - \int \phi(x^5) \, dt \right) \]