\( \int\displaystyle 5^x\,dx \) is
(where C is an arbitrary constant)
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( \displaystyle\frac{5^x}{\log_e 5}+\mathrm{C} \)
By the standard result \( \int a^x\,dx=\frac{a^x}{\log_e a}+C \),
\( \displaystyle \int 5^x\,dx=\frac{5^x}{\log_e 5}+C \)
Why the other options are incorrect
Option (1) \( 5^x\log_e 5+\mathrm{C} \) — Incorrect. \( 5^x\log_e 5 \) is the derivative of \( 5^x \), not its integral.
Option (2) \( 5^x\log_e x+\mathrm{C} \) — Incorrect. \( 5^x \) must be divided by \( \log_e 5 \), not multiplied by \( \log_e x \).
Option (3) \( \displaystyle\frac{5^x}{\log_e x}+\mathrm{C} \) — Incorrect. The divisor is \( \log_e 5 \), the log of the base, not \( \log_e x \).
NCERT (2026-27 edition):
Integration as an Inverse Process of Differentiation
