Question

If \(\int \sqrt{2}\sqrt{1+sinx}\,dx = -4cos(ax+b)+c\), then the value of \(a,b\) respectively are...

• A. \(\frac{1}{2},\frac{π}{2}\)
• B. \(\frac{1}{2},\frac{π}{4}\)
• C. \(\frac{x}{2},\frac{π}{4}\)
• D. \(1,\frac{π}{2}\)

Hint:

Write 1 + sin x as 2 sin^2(x/2 + pi/4).

Answer 1

The Correct Option is B

Step 1: Differentiate the answer:
Check the options by differentiating $-4\cos(ax+b)$: the derivative is $4a\sin(ax+b)$.

Step 2: Match:
We need $4a\sin(ax+b)=\sqrt2\sqrt{1+\sin x}$. With $a=\tfrac12$, $b=\tfrac\pi4$: $2\sin(\tfrac x2+\tfrac\pi4)$.

Step 3: Square check:
Its square is $4\sin^2(\ldots)=2(1+\sin x)$, and the target squared is $2(1+\sin x)$. They agree.

Final Answer:
Square both sides to confirm the match. \[ \boxed{B} \]