`int(sqrt(sinx))cosxdx=?`

Options

A. `(2)/(3)(cosx)^(3/2)+C`
B. `(3)/(2)(cosx)^(3/2)+C`
C. `(2)/(3)(sinx)^(3/2)+C`
D. `(3)/(2)(sinx)^(3/2)+C`

Explanation

To solve the integral \( \int \sqrt{\sin x} \cos x \, dx \), we can follow these steps:

### Step 1: Substitution
Let \( t = \sin x \). Then, we differentiate \( t \) with respect to \( x \):
\[ dt = \cos x \, dx \]
This means that \( \cos x \, dx = dt \).

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

### Step 3: Integrate
Next, we integrate \( \sqrt{t} \):
\[ \int \sqrt{t} \, dt = \int t^{1/2} \, dt = \frac{t^{3/2}}{3/2} + C = \frac{2}{3} t^{3/2} + C \]

### Step 4: Substitute Back
Now, we substitute back \( t = \sin x \):
\[ \frac{2}{3} (\sin x)^{3/2} + C \]

### Final Answer
Thus, the integral \( \int \sqrt{\sin x} \cos x \, dx \) is:
\[ \frac{2}{3} \sin^{3/2} x + C \]

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