Evaluate the integral.
\[
\int_{0}^{\pi / 6} 3 \sec ^{2} x d x
\]

Answer

Expert–verified

Hide Steps

Answer

Final Answer: \(\boxed{\sqrt{3}}\)

Steps

Step 1 ：The integral of \(\sec^2(x)\) is \(\tan(x)\). Therefore, the integral of \(3\sec^2(x)\) is \(3\tan(x)\).

Step 2 ：We need to evaluate this from 0 to \(\frac{\pi}{6}\).

Step 3 ：The integral of \(3\sec^2(x)\) from 0 to \(\frac{\pi}{6}\) is \(\sqrt{3}\).

Step 4 ：Final Answer: \(\boxed{\sqrt{3}}\)