You are examine these two statements carefully and select and answer
Assertion (A) : `int_0^pi sin^7 x dx =2 int_0^(pi//2) sin^7 xdx`
Reason ( R):` sin^7x` is an odd function

Options

A. Both A and R are individually true and R is the correct
B. Both A and R are individually true but R is not the correct explanation A
C. A is true but R is false
D. A is false but R is true

Answer

To solve the problem, we need to analyze both the Assertion (A) and the Reason (R) provided in the question.

### Step 1: Analyze the Assertion (A)
The assertion states:
\[ \int_0^\pi \sin^7 x \, dx = 2 \int_0^{\frac{\pi}{2}} \sin^7 x \, dx \]
To verify this, we can use the property of definite integrals. The integral of a function over a symmetric interval can be expressed in terms of the integral over half the interval.

### Step 2: Use the Symmetry of the Function
The function \(\sin^7 x\) is symmetric about \(\frac{\pi}{2}\) in the interval \([0, \pi]\). This means:
\[ \sin^7(\pi - x) = \sin^7 x \]
for \(x \in [0, \pi]\). Thus, we can say:
\[ \int_0^\pi \sin^7 x \, dx = \int_0^{\frac{\pi}{2}} \sin^7 x \, dx + \int_{\frac{\pi}{2}}^\pi \sin^7 x \, dx \]
By changing the variable in the second integral, let \(u = \pi - x\), then \(du = -dx\), and when \(x = \frac{\pi}{2}\), \(u = \frac{\pi}{2}\) and when \(x = \pi\), \(u = 0\). Thus,
\[ \int_{\frac{\pi}{2}}^\pi \sin^7 x \, dx = \int_0^{\frac{\pi}{2}} \sin^7(\pi - u) \, (-du) = \int_0^{\frac{\pi}{2}} \sin^7 u \, du \]
This implies:
\[ \int_{\frac{\pi}{2}}^\pi \sin^7 x \, dx = \int_0^{\frac{\pi}{2}} \sin^7 x \, dx \]
Thus, we can conclude:
\[ \int_0^\pi \sin^7 x \, dx = 2 \int_0^{\frac{\pi}{2}} \sin^7 x \, dx \]
This confirms that Assertion (A) is correct.

### Step 3: Analyze the Reason (R)
The reason states:
\[ \sin^7 x \text{ is an odd function} \]
To determine if this is true, we need to check the definition of an odd function. A function \(f(x)\) is odd if:
\[ f(-x) = -f(x) \]
However, \(\sin^7 x\) is not an odd function because:
\[ \sin^7(-x) = (-\sin x)^7 = -\sin^7 x \]
This means that \(\sin^7 x\) is indeed an odd function about the origin, but we are integrating over the interval \([0, \pi]\), which does not reflect the properties of odd functions in the same way.

### Conclusion
- Assertion (A) is correct.
- Reason (R) is true but does not correctly explain the assertion.

### Final Answer
The correct relationship is that Assertion (A) is true, and Reason (R) is true but does not provide the correct explanation for Assertion (A).