Question

$\int \sin \frac{y}{2} \, dy =$

• A. $2 \cos \frac{y}{2} + c$
• B. $2 \sin x/2 + c$
• C. $2 \cos 2y + c$
• D. $-2 \cos \frac{y}{2} + c$

Hint:

Always remember: "Integral of Sine is Negative Cosine." A quick check is to differentiate your answer. The derivative of $-2 \cos(y/2)$ is $-2(-\sin(y/2) \cdot 1/2) = \sin(y/2)$, confirming the result.

Answer 1

The Correct Option is D

This integral is solved using the basic trigonometric integration rule combined with the reverse of the chain rule (u-substitution). 1. Standard Integration Rule: Recall that $\int \sin(ax) \, dx = -\frac{1}{a} \cos(ax) + c$.

2. Identification of Parameters: In the expression $\sin \frac{y}{2}$, the variable is $y$ and the coefficient $a$ is $\frac{1}{2}$.

3. Calculation: Applying the rule: $$\int \sin \frac{y}{2} \, dy = -\frac{1}{1/2} \cos \frac{y}{2} + c$$ $$\text{Integral} = -2 \cos \frac{y}{2} + c$$ Option (D) is the only mathematically sound answer. Note that Option (B) incorrectly changes the variable to $x$.