Indefinite Integration
Standard Forms
Grade 12
Question:
<p>The value of $\displaystyle \int \frac{dx}{\sqrt{a^2-x^2}}$ is equal to</p>
<p>(A) $\frac{1}{2}\sin^{-1}\left(\frac{x}{a}\right) + \frac{1}{2}\ln|x+\sqrt{a^2-x^2}| + C$</p>
<p>(B) $\sin^{-1}\left(\frac{x}{a}\right) + \frac{1}{2}\ln|x+\sqrt{a^2-x^2}| + C$</p>
<p>(C) Standard form</p>
<p>(D) none of these</p>
Step-by-Step Solution
Key Concept: Recognize this as a standard inverse trigonometric integral
<p>This is a standard integral form. The antiderivative of $\frac{1}{\sqrt{a^2-x^2}}$ is $\sin^{-1}\left(\frac{x}{a}\right) + C$.</p>
Correct Answer: A