\( \int_{-2}^{2}\displaystyle\frac{x^4}{1+5^x}\,dx \) is equal to
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( \displaystyle\frac{32}{5} \)
Since the interval is symmetric, replacing x by −x does not change the integral:
\( \displaystyle I=\int_{-2}^{2}\frac{x^4}{1+5^{-x}}\,dx \)
\( \displaystyle =\int_{-2}^{2}\frac{5^x\,x^4}{5^x+1}\,dx \)
Adding the two forms of I:
\( \displaystyle 2I=\int_{-2}^{2}x^4\,dx \)
\( \displaystyle =\frac{64}{5} \)
\( \displaystyle I=\frac{32}{5} \)
Why the other options are incorrect
Option (1) 0 — Incorrect. The integrand is positive for every x ≠ 0, so the integral cannot be 0.
Option (2) \( \displaystyle\frac{4}{3} \) — Incorrect. The value is \( \frac{32}{5} \), not \( \frac{4}{3} \).
Option (4) \( \displaystyle\frac{64}{5} \) — Incorrect. This is 2I; it must be halved.