Question

evaluate the following integral using trigonometric substitution
int_{1}^{3}sqrt{4 - x^{2}}dx
what substitution will be the most helpful for evaluating this integral?
a. (x = 2sec\theta)
b. (x = 2sin\theta)
c. (x = 2\tan\theta)

Explanation:

Step1: Recall trig - sub rules

For an integral of the form $\int\sqrt{a^{2}-x^{2}}dx$, the substitution $x = a\sin\theta$ is useful. Here $a = 2$ and the integral is $\int_{1}^{2}\sqrt{4 - x^{2}}dx$.

Step2: Analyze the substitution forms

If $x=2\sin\theta$, then $dx = 2\cos\theta d\theta$ and $\sqrt{4 - x^{2}}=\sqrt{4 - 4\sin^{2}\theta}=2\cos\theta$.

Answer:

B. $x = 2\sin\theta$