The value of `intsin^(3)x sqrt(cosx)dx` is equal to (where, c is the constant of integration)

Options

A. `(2)/(3)(cosx)^((3)/(2))+(2)/(7)(cosx)^((7)/(2))+c`
B. `-(2)/(3)(cosx)^((3)/(2))+(2)/(7)(cosx)^((7)/(2))+c`
C. `(-(2)/(3))(cosx)^(3)+(2)/(5)(cosx^((5)/(2)))+c`
D. `(3)/(2)(cosx)^((3)/(2))+(5)/(2)(cosx)^((7)/(2))+c`

Answer

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

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

### Step 2: Use the Pythagorean Identity
Using the identity \( \sin^2 x = 1 - \cos^2 x \), we can rewrite the integral:
\[ \int (1 - \cos^2 x) \sin x \sqrt{\cos x} \, dx \]

### Step 3: Substitution
Let \( \cos x = t \). Then, the derivative \( -\sin x \, dx = dt \) or \( \sin x \, dx = -dt \). The limits of integration will change accordingly, but since we are looking for the indefinite integral, we can ignore the limits for now:
\[ \int (1 - t^2) \sqrt{t} (-dt) \]

### Step 4: Simplify the Integral
This becomes:
\[ -\int (1 - t^2) t^{1/2} \, dt = -\int (t^{1/2} - t^{5/2}) \, dt \]

### Step 5: Integrate Term by Term
Now we can integrate term by term:
\[ -\left( \int t^{1/2} \, dt - \int t^{5/2} \, dt \right) \]
Calculating the integrals:
\[ -\left( \frac{t^{3/2}}{3/2} - \frac{t^{7/2}}{7/2} \right) = -\left( \frac{2}{3} t^{3/2} - \frac{2}{7} t^{7/2} \right) \]

### Step 6: Substitute Back
Substituting back \( t = \cos x \):
\[ -\left( \frac{2}{3} \cos^{3/2} x - \frac{2}{7} \cos^{7/2} x \right) + C \]
This simplifies to:
\[ \frac{2}{3} \cos^{3/2} x - \frac{2}{7} \cos^{7/2} x + C \]

### Final Answer
Thus, the value of the integral is:
\[ \int \sin^3 x \sqrt{\cos x} \, dx = \frac{2}{3} \cos^{3/2} x - \frac{2}{7} \cos^{7/2} x + C \]

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