The value of \(\int_{0}^{1}\displaystyle x(1 - x)^{2022}\,dx\) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{1}{2023 \times 2024}\)
Expanding \((1 - x)^{2022}\) is not practical. By the property \(\int_0^a f(x)\,dx = \int_0^a f(a - x)\,dx\) with a = 1, replace x by 1 − x; the large power moves onto x:
\( \displaystyle \int_{0}^{1}x(1 - x)^{2022}\,dx = \int_{0}^{1}(1 - x)\,x^{2022}\,dx \)
\( \displaystyle = \int_{0}^{1}\left(x^{2022} - x^{2023}\right)dx \)
\( \displaystyle = \frac{1}{2023} - \frac{1}{2024} \)
\( \displaystyle = \frac{2024 - 2023}{2023 \times 2024} \)
\( \displaystyle = \frac{1}{2023 \times 2024} \)