What is `intsin^(3)xcosxdx` equal to ?

Options

A. `cos^(4)x+c`
B. `sin^(4)x+c`
C. `((1-sin^(2)x)^(2))/(4)+c`
D. `((1-cos^(2)x)^(2))/(4)+c`

Answer

To solve the integral \( \int \sin^3 x \cos x \, dx \), we can use substitution and integration techniques. Here’s a step-by-step solution:

### Step 1: Use Substitution
Let \( u = \sin x \). Then, the derivative \( du = \cos x \, dx \). This means that \( dx = \frac{du}{\cos x} \).

### Step 2: Rewrite the Integral
Substituting \( u \) into the integral, we have:
\[ \int \sin^3 x \cos x \, dx = \int u^3 \, du \]

### Step 3: Integrate
Now, we can integrate \( u^3 \):
\[ \int u^3 \, du = \frac{u^4}{4} + C \]

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

### Step 5: Final Answer
Thus, the final result of the integral is:
\[ \int \sin^3 x \cos x \, dx = \frac{\sin^4 x}{4} + C \]