`int_(0)^(pi//2)sin^(5)x dx=`

Options

A. `(15)/(8)`
B. `(8)/(15)`
C. `(8pi)/(15)`
D. `(15pi)/(8)`

Answer

To solve the integral \( \int_{0}^{\frac{\pi}{2}} \sin^5 x \, dx \), we can follow these steps:

### Step 1: Rewrite the integral
We start by rewriting \( \sin^5 x \) in a more manageable form:
\[ \sin^5 x = \sin^4 x \cdot \sin x \]
Thus, we can express the integral as:
\[ \int_{0}^{\frac{\pi}{2}} \sin^5 x \, dx = \int_{0}^{\frac{\pi}{2}} \sin^4 x \cdot \sin x \, dx \]

### Step 2: Use the identity for \( \sin^2 x \)
We know that \( \sin^2 x = 1 - \cos^2 x \). Therefore, we can write:
\[ \sin^4 x = (\sin^2 x)^2 = (1 - \cos^2 x)^2 \]
Expanding this gives:
\[ \sin^4 x = 1 - 2\cos^2 x + \cos^4 x \]
Now, substituting this back into the integral, we have:
\[ \int_{0}^{\frac{\pi}{2}} \sin^5 x \, dx = \int_{0}^{\frac{\pi}{2}} (1 - 2\cos^2 x + \cos^4 x) \sin x \, dx \]

### Step 3: Split the integral
We can split the integral into three parts:
\[ \int_{0}^{\frac{\pi}{2}} \sin x \, dx - 2 \int_{0}^{\frac{\pi}{2}} \cos^2 x \sin x \, dx + \int_{0}^{\frac{\pi}{2}} \cos^4 x \sin x \, dx \]

### Step 4: Evaluate the first integral
The first integral is straightforward:
\[ \int_{0}^{\frac{\pi}{2}} \sin x \, dx = -\cos x \bigg|_{0}^{\frac{\pi}{2}} = -\cos\left(\frac{\pi}{2}\right) + \cos(0) = 0 + 1 = 1 \]

### Step 5: Evaluate the second integral
For the second integral, we use the substitution \( t = \cos x \), hence \( dt = -\sin x \, dx \). The limits change from \( x = 0 \) to \( x = \frac{\pi}{2} \) which corresponds to \( t = 1 \) to \( t = 0 \):
\[ \int_{0}^{\frac{\pi}{2}} \cos^2 x \sin x \, dx = \int_{1}^{0} t^2 (-dt) = \int_{0}^{1} t^2 \, dt = \frac{t^3}{3} \bigg|_{0}^{1} = \frac{1}{3} \]
Thus,
\[ -2 \int_{0}^{\frac{\pi}{2}} \cos^2 x \sin x \, dx = -2 \cdot \frac{1}{3} = -\frac{2}{3} \]

### Step 6: Evaluate the third integral
For the third integral, we again use the substitution \( t = \cos x \):
\[ \int_{0}^{\frac{\pi}{2}} \cos^4 x \sin x \, dx = \int_{1}^{0} t^4 (-dt) = \int_{0}^{1} t^4 \, dt = \frac{t^5}{5} \bigg|_{0}^{1} = \frac{1}{5} \]

### Step 7: Combine all parts
Now we can combine all the parts:
\[ \int_{0}^{\frac{\pi}{2}} \sin^5 x \, dx = 1 - \frac{2}{3} + \frac{1}{5} \]
To combine these fractions, we find a common denominator, which is 15:
\[ = \frac{15}{15} - \frac{10}{15} + \frac{3}{15} = \frac{15 - 10 + 3}{15} = \frac{8}{15} \]

### Final Answer
Thus, the value of the integral is:
\[ \int_{0}^{\frac{\pi}{2}} \sin^5 x \, dx = \frac{8}{15} \]

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