Problem 18 - Essential Calculus

Question

Evaluate the integral. $$ \int_{\sqrt{3}}^{3} \frac{1}{9+x^{2}} d x $$

Step-by-Step Solution

Answer

The value of the given integral is \( \frac{\pi}{12} \)

Step 1: Identify the integral type

We can see that the integral is of the form \( \frac{1}{a^2+x^2} \), which can be solved using the arctan formula.

Step 2: Apply the arctan formula to the integral

Applying the arctan formula we have that \( \int \frac{1}{9+x^{2}} dx = \frac{1}{3} arctan( \frac{x}{3} ) + C \)

Step 3: Evaluate the integral

Evaluate the integral at the upper and lower limits. Plug in the upper limit \(3\) and the lower limit \(\sqrt{3}\) into the antiderivative and subtract the results.\n \( \frac{1}{3} [arctan(1) - arctan( \frac{\sqrt{3}}{3} )] = \frac{\pi}{12} \)