Find `intsin^(3)x cos x dx`.

Answer

To solve the integral \( \int \sin^3(x) \cos(x) \, dx \), we can follow these steps:

### Step 1: Use a Substitution
Let \( t = \sin(x) \). Then, the derivative \( dt = \cos(x) \, dx \).

### Step 2: Change the Integral
Now, we can rewrite the integral in terms of \( t \):
\[ \int \sin^3(x) \cos(x) \, dx = \int t^3 \, dt \]

### Step 3: Integrate
Now, we can integrate \( t^3 \):
\[ \int t^3 \, dt = \frac{t^4}{4} + C \]
where \( C \) is the constant of integration.

### Step 4: Substitute Back
Now, we substitute back \( t = \sin(x) \):
\[ \frac{t^4}{4} + C = \frac{\sin^4(x)}{4} + C \]

### Final Answer
Thus, the integral \( \int \sin^3(x) \cos(x) \, dx \) is:
\[ \frac{\sin^4(x)}{4} + C \]

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