`int_(0)^(pi//2)sin^(3)x dx=`

Options

A. `(2)/(3)`
B. `(3)/(2)`
C. `-(2)/(3)`
D. `-(3)/(2)`

Answer

To solve the integral \( \int_{0}^{\frac{\pi}{2}} \sin^3 x \, dx \), we can follow these steps:

### Step 1: Rewrite the Integral
We can express \( \sin^3 x \) as \( \sin^2 x \cdot \sin x \). Thus, we rewrite the integral:
\[ \int_{0}^{\frac{\pi}{2}} \sin^3 x \, dx = \int_{0}^{\frac{\pi}{2}} \sin^2 x \cdot \sin x \, dx \]

### Step 2: Use the Pythagorean Identity
Using the identity \( \sin^2 x = 1 - \cos^2 x \), we can substitute:
\[ \int_{0}^{\frac{\pi}{2}} \sin^2 x \cdot \sin x \, dx = \int_{0}^{\frac{\pi}{2}} (1 - \cos^2 x) \sin x \, dx \]

### Step 3: Split the Integral
Now, we can split the integral into two parts:
\[ \int_{0}^{\frac{\pi}{2}} (1 - \cos^2 x) \sin x \, dx = \int_{0}^{\frac{\pi}{2}} \sin x \, dx - \int_{0}^{\frac{\pi}{2}} \cos^2 x \sin x \, dx \]

### Step 4: Evaluate the First Integral
The first integral \( \int_{0}^{\frac{\pi}{2}} \sin x \, dx \) can be evaluated directly:
\[ \int \sin x \, dx = -\cos x \quad \text{so} \quad \int_{0}^{\frac{\pi}{2}} \sin x \, dx = [-\cos x]_{0}^{\frac{\pi}{2}} = -\cos\left(\frac{\pi}{2}\right) + \cos(0) = 0 + 1 = 1 \]

### Step 5: Evaluate the Second Integral
For the second integral \( \int_{0}^{\frac{\pi}{2}} \cos^2 x \sin x \, dx \), we can use substitution. Let \( t = \cos x \), then \( dt = -\sin x \, dx \). Changing the limits:
- When \( x = 0 \), \( t = \cos(0) = 1 \)
- When \( x = \frac{\pi}{2} \), \( t = \cos\left(\frac{\pi}{2}\right) = 0 \)

Thus, we have:
\[ \int_{0}^{\frac{\pi}{2}} \cos^2 x \sin x \, dx = -\int_{1}^{0} t^2 \, dt = \int_{0}^{1} t^2 \, dt \]
Now, we can evaluate this integral:
\[ \int t^2 \, dt = \frac{t^3}{3} \quad \text{so} \quad \int_{0}^{1} t^2 \, dt = \left[\frac{t^3}{3}\right]_{0}^{1} = \frac{1^3}{3} - \frac{0^3}{3} = \frac{1}{3} \]

### Step 6: Combine the Results
Now we can combine the results from the two integrals:
\[ \int_{0}^{\frac{\pi}{2}} \sin^3 x \, dx = 1 - \frac{1}{3} = \frac{3}{3} - \frac{1}{3} = \frac{2}{3} \]

Thus, the final result is:
\[ \int_{0}^{\frac{\pi}{2}} \sin^3 x \, dx = \frac{2}{3} \]