Discussion
The integral \( \displaystyle\int \frac{dx}{a^{2} + x^{2}} \) equals
Answer: B
Substituting \( x = a\tan\theta \) gives \( dx = a\sec^{2}\theta\,d\theta \) and turns the denominator into \( a^{2}\sec^{2}\theta \), leaving
\[ \int \frac{a\sec^{2}\theta\,d\theta}{a^{2}\sec^{2}\theta} \;=\; \frac{\theta}{a} \;=\; \frac{1}{a}\tan^{-1}\frac{x}{a} + C. \]
The inverse sine belongs to \( 1/\sqrt{a^{2}-x^{2}} \), and the logarithm to \( 1/(a^{2}-x^{2}) \)
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