The value of `int_(0)^(pi)abscosx^3dx` is

Options

A. `2//3`
B. 0
C. `-4//3`
D. `4//3`

Answer

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

### Step 1: Analyze the function
The function \( |\cos x| \) is non-negative and periodic. Over the interval \( [0, \pi] \), \( \cos x \) is non-negative from \( 0 \) to \( \frac{\pi}{2} \) and non-positive from \( \frac{\pi}{2} \) to \( \pi \). Therefore, we can split the integral at \( \frac{\pi}{2} \).

### Step 2: Split the integral
We can express the integral as:
\[ \int_{0}^{\pi} |\cos x|^3 \, dx = \int_{0}^{\frac{\pi}{2}} \cos^3 x \, dx + \int_{\frac{\pi}{2}}^{\pi} |\cos x|^3 \, dx \]
Since \( \cos x \) is non-negative in the first interval and non-positive in the second, we have:
\[ \int_{\frac{\pi}{2}}^{\pi} |\cos x|^3 \, dx = \int_{\frac{\pi}{2}}^{\pi} (-\cos x)^3 \, dx = -\int_{\frac{\pi}{2}}^{\pi} \cos^3 x \, dx \]

### Step 3: Combine the integrals
Thus, we can write:
\[ \int_{0}^{\pi} |\cos x|^3 \, dx = \int_{0}^{\frac{\pi}{2}} \cos^3 x \, dx - \int_{\frac{\pi}{2}}^{\pi} \cos^3 x \, dx \]
This simplifies to:
\[ \int_{0}^{\pi} |\cos x|^3 \, dx = 2 \int_{0}^{\frac{\pi}{2}} \cos^3 x \, dx \]

### Step 4: Use the reduction formula
To evaluate \( \int_{0}^{\frac{\pi}{2}} \cos^3 x \, dx \), we can use the reduction formula:
\[ \int_{0}^{\frac{\pi}{2}} \cos^n x \, dx = \frac{n-1}{n} \int_{0}^{\frac{\pi}{2}} \cos^{n-2} x \, dx \]
For \( n = 3 \):
\[ \int_{0}^{\frac{\pi}{2}} \cos^3 x \, dx = \frac{2}{3} \int_{0}^{\frac{\pi}{2}} \cos x \, dx \]
And we know:
\[ \int_{0}^{\frac{\pi}{2}} \cos x \, dx = 1 \]
Thus:
\[ \int_{0}^{\frac{\pi}{2}} \cos^3 x \, dx = \frac{2}{3} \cdot 1 = \frac{2}{3} \]

### Step 5: Final calculation
Now substituting back:
\[ \int_{0}^{\pi} |\cos x|^3 \, dx = 2 \cdot \frac{2}{3} = \frac{4}{3} \]

### Final Answer
The value of \( \int_{0}^{\pi} |\cos x|^3 \, dx \) is \( \frac{4}{3} \).

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