`int cos^3 xe^("log" (sin x)) dx` is equal to

Options

A. `- (sin^4 x)/(4) + c`
B. `- ("cos"^4 x)/(4) + c`
C. `(e^(sin x))/(4) +c `
D. none of these

Answer

To solve the integral \( \int \cos^3 x \, e^{\log(\sin x)} \, dx \), we will follow these steps:

### Step 1: Simplify the integrand
We know that \( e^{\log(a)} = a \). Therefore, we can simplify the integrand:
\[ e^{\log(\sin x)} = \sin x \]
Thus, the integral becomes:
\[ \int \cos^3 x \sin x \, dx \]

### Step 2: Use substitution
Let us use the substitution:
\[ t = \cos x \]
Then, the derivative of \( t \) with respect to \( x \) is:
\[ dt = -\sin x \, dx \quad \Rightarrow \quad \sin x \, dx = -dt \]
Now, we can rewrite the integral in terms of \( t \):
\[ \int \cos^3 x \sin x \, dx = \int t^3 (-dt) = -\int t^3 \, dt \]

### Step 3: Integrate
Now we can integrate \( -\int t^3 \, dt \):
\[ -\int t^3 \, dt = -\left(\frac{t^4}{4}\right) + C = -\frac{t^4}{4} + C \]

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

### Final Answer
Thus, the final answer is:
\[ \int \cos^3 x \, e^{\log(\sin x)} \, dx = -\frac{\cos^4 x}{4} + C \]

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