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COMEDK 2025 Afternoon Solved Paper

Question : 147 of 180

Marks: +1, -0

\int \frac{\sin x+\cos x}{\sqrt{1+2 \sin x \cos x}} d x=\varphi(x)+C

\varphi(x)=

Solution:

Concept:

Simplify the integrand using the trigonometric identity

1 + 2\sin x \cos x = (\sin x + \cos x)^2

Explanation:

Start by rewriting the expression under the square root.

1 + 2\sin x \cos x = \sin^2 x + \cos^2 x + 2\sin x \cos x = (\sin x + \cos x)^2

Assuming

\sin x + \cos x > 0

, the square root simplifies to

\sqrt{(\sin x + \cos x)^2} = \sin x + \cos x

The integrand becomes

\frac{\sin x + \cos x}{\sin x + \cos x} = 1

Therefore, the integral is

\int 1 \, dx = x + C

Hence

\varphi(x) = x

Answer:

x

(Option B)

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