Question

\(\int_{0}^{\pi}x \sin^4 x \cos^6 x dx = \)

• A. \(\frac{3\pi^2}{512}\)
• B. \(\frac{3\pi^2}{256}\)
• C. \(\frac{\pi^2}{256}\)
• D. \(\frac{\pi^2}{512}\)

Hint:

Use the properties of definite integrals to simplify the expression and then use the formula for \int_{0}^{\pi/2} \sin^m x \cos^n x dx.

Answer 1

The Correct Option is A

Step 1: Use the property \int_{0^{a} f(x) dx = \int_{0}^{a} f(a-x) dx.}
Let I = \int_{0}^{\pi} x \sin^4 x \cos^6 x dx.
Using the property, I = \int_{0}^{\pi} (\pi - x) \sin^4 (\pi - x) \cos^6 (\pi - x) dx.
I = \int_{0}^{\pi} (\pi - x) \sin^4 x (-\cos x)^6 dx.
I = \int_{0}^{\pi} (\pi - x) \sin^4 x \cos^6 x dx.
I = \int_{0}^{\pi} \pi \sin^4 x \cos^6 x dx - \int_{0}^{\pi} x \sin^4 x \cos^6 x dx.
I = \pi \int_{0}^{\pi} \sin^4 x \cos^6 x dx - I.
2I = \pi \int_{0}^{\pi} \sin^4 x \cos^6 x dx.
I = \frac{\pi}{2} \int_{0}^{\pi} \sin^4 x \cos^6 x dx. Step 2: Use the property \int_{0^{2a} f(x) dx = 2 \int_{0}^{a} f(x) dx if f(2a-x) = f(x).}
Since \sin^4 (\pi - x) \cos^6 (\pi - x) = \sin^4 x \cos^6 x, we have: I = \frac{\pi}{2} \cdot 2 \int_{0}^{\pi/2} \sin^4 x \cos^6 x dx.
I = \pi \int_{0}^{\pi/2} \sin^4 x \cos^6 x dx. Step 3: Use the formula for \int_{0^{\pi/2} \sin^m x \cos^n x dx.}
\int_{0}^{\pi/2} \sin^m x \cos^n x dx = \frac{(m-1)(m-3)\cdots 1 (n-1)(n-3)\cdots 1}{(m+n)(m+n-2)\cdots 2} \cdot \frac{\pi}{2} if both m and n are even.
Here, m = 4 and n = 6. I = \pi \cdot \frac{(4-1)(4-3)(6-1)(6-3)(6-5)}{(4+6)(4+6-2)(4+6-4)(4+6-6)(4+6-8)} \cdot \frac{\pi}{2}
I = \pi \cdot \frac{3 \cdot 1 \cdot 5 \cdot 3 \cdot 1}{10 \cdot 8 \cdot 6 \cdot 4 \cdot 2} \cdot \frac{\pi}{2}
I = \pi \cdot \frac{45}{3840} \cdot \frac{\pi}{2} = \pi \cdot \frac{3}{256} \cdot \frac{\pi}{2} = \frac{3\pi^2}{512}. Therefore, the integral is \frac{3\pi^2}{512}.