Question

Evaluate the integral: \[ \int \sqrt{\cos x} \cdot \sin x \, dx = ? \]

• A. \( \frac{2}{3} (\cos x)^{3/2} + c \)
• B. \( -\frac{2}{3} (\cos x)^{3/2} + c \)
• C. \( (\cos x)^{3/2} + c \)
• D. \( -(\cos x)^{3/2} + c \)

Hint:

When integrating products like \( \sqrt{\cos x} \sin x \), substitution \( t = \cos x \) simplifies the integral.

Answer 1

The Correct Option is B

Let: \[ I = \int \sqrt{\cos x} \cdot \sin x \, dx = \int (\cos x)^{1/2} \sin x \, dx \] Use substitution: \[ t = \cos x \implies dt = -\sin x \, dx \implies -dt = \sin x \, dx \] So, \[ I = \int t^{1/2} (-dt) = -\int t^{1/2} dt = -\frac{2}{3} t^{3/2} + c = -\frac{2}{3} (\cos x)^{3/2} + c \]