Use the trigonometric substitution u = a sin θ, where

Question

Use the trigonometric substitution u = a sin θ, where - π/2 < θ < π/2 and a > 0 to simplify the expression  . ​
A)​a csc θ 
B)​a sin θ 
C)​a tan θ 
D)​a cos θ 
E)​- a cos θ

Quizplus · Accepted Answer

Using the trigonometric substitution u = a sin θ, we can rewrite the expression as: 
$$\sqrt{a^2-u^2}=\sqrt{a^2-a^2\sin^2	heta}=a\cos	heta$$
Therefore, the simplified expression is:
$$\int \sqrt{a^2-u^2} du = \int a\cos	heta d	heta = a\sin	heta + C$$
Since $\sin 	heta = \frac{u}{a}$, we have:
$$a\sin	heta + C = \boxed{D)~a\cos	heta}$$