Question

\(\int _{0}^{\frac{\pi }{2}}(1+\sin x)^{3}dx=\)

• A. \(0\)
• B. \(1\)
• C. \(\frac{\pi }{2}-\frac{2}{3}\)
• D. \(\frac{5\pi }{4}+\frac{11}{3}\)

Hint:

Expand the cube and integrate each term using the standard integrals of sine and sine squared and sine cubed.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
Expand \((1+\sin x)^{3}=1+3\sin x+3\sin^{2}x+\sin^{3}x\) and integrate term by term over \([0,\frac{\pi}{2}]\).

Step 2: Easy terms
\(\int_{0}^{\pi/2}1\,dx=\frac{\pi}{2}\) and \(\int_{0}^{\pi/2}3\sin x\,dx=3\).

Step 3: Higher powers
\(\int_{0}^{\pi/2}\sin^{2}x\,dx=\frac{\pi}{4}\), so \(3\sin^{2}x\) gives \(\frac{3\pi}{4}\).
\(\int_{0}^{\pi/2}\sin^{3}x\,dx=\frac{2}{3}\).

Step 4: Add
\(\frac{\pi}{2}+3+\frac{3\pi}{4}+\frac{2}{3}=\frac{5\pi}{4}+\frac{11}{3}\).

Final Answer:
The value is 5π/4 + 11/3. \[ \boxed{\text{(D) }\dfrac{5\pi}{4}+\dfrac{11}{3}} \]