Evaluate :
`int x^(5)sin x^(3)dx`

Answer

To evaluate the integral \( I = \int x^5 \sin(x^3) \, dx \), we can follow these steps:

### Step 1: Rewrite the Integral
We can express \( x^5 \) as \( x^2 \cdot x^3 \):
\[ I = \int x^2 \cdot x^3 \sin(x^3) \, dx \]

**Hint:** Look for a substitution that simplifies the integral, especially when you see a function and its derivative.

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

**Hint:** When substituting, always remember to change the variable in both the differential and the integrand.

### Step 3: Change the Integral
Substituting \( t \) into the integral, we have:
\[ I = \int x^2 \sin(t) \cdot \frac{dt}{3} = \frac{1}{3} \int \sin(t) \, dt \]

**Hint:** Simplifying the integral can make it easier to integrate.

### Step 4: Integrate
Now we can integrate \( \sin(t) \):
\[ I = \frac{1}{3} \left( -\cos(t) \right) + C = -\frac{1}{3} \cos(t) + C \]

**Hint:** Remember to include the constant of integration after performing the integral.

### Step 5: Back Substitute
Now, we substitute back \( t = x^3 \):
\[ I = -\frac{1}{3} \cos(x^3) + C \]

**Hint:** Always revert to the original variable to express the final answer.

### Final Answer
Thus, the evaluated integral is:
\[ \int x^5 \sin(x^3) \, dx = -\frac{1}{3} \cos(x^3) + C \]