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: Understanding the Concept:
We write \(1+\sin x\) as a perfect square using half angles.

Step 2: Rewrite:
\[ 1+\sin x=\left(\sin\tfrac x2+\cos\tfrac x2\right)^2=2\sin^2\left(\tfrac x2+\tfrac\pi4\right) \]
So \(\sqrt{1+\sin x}=\sqrt2\,\sin\left(\tfrac x2+\tfrac\pi4\right)\) (taking the positive branch).

Step 3: Multiply by sqrt 2:
\(\sqrt2\sqrt{1+\sin x}=2\sin\left(\tfrac x2+\tfrac\pi4\right)\).

Step 4: Integrate:
\[ \int2\sin\left(\tfrac x2+\tfrac\pi4\right)dx=-4\cos\left(\tfrac x2+\tfrac\pi4\right)+c \]

Step 5: Compare:
\(a=\dfrac12\) and \(b=\dfrac\pi4\), option (B).

Final Answer:
The integral is -4 cos(x/2 + pi/4) + c, so a = 1/2 and b = pi/4. \[ \boxed{a=\tfrac12,\ b=\tfrac\pi4} \]