For \(x \gt 0\), \(\int 9^{\log_3 x}\,dx\) is equal to (where C is a constant of integration)
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{x^3}{3} + C\)
Since \(9 = 3^2\) and \(3^{\log_3 x} = x\),
\(9^{\log_3 x} = \left(3^{\log_3 x}\right)^2\)
\(= x^2\)
So the integral is a simple power:
\( \displaystyle \int x^2\,dx = \frac{x^3}{3} + C \)