Definite Integration Question 2
Question 2 - 24 January - Shift 1
The value of $\frac{8}{\pi} \int_0^{\frac{\pi}{2}} \frac{(\cos x)^{2023} 90}{(\sin x)^{2023}+(\cos x)^{2023}} d x$ is
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Answer: (2)
Solution:
Formula: Properties of definite integral , Standard Formula
$I=\frac{8}{\pi} \int_0^{\frac{\pi}{2}} \frac{(\cos x)^{2023}}{(\sin x)^{2023}+(\cos x)^{2023}} d x \ldots (1) $
Using $\int_0^{a} f(x) d x=\int_0^{a} f(a-x) d x$
$I=\frac{8}{\pi} \int_0^{\frac{\pi}{2}} \frac{(\sin x)^{2023}}{(\sin x)^{2023}+(\cos x)^{2023}} d x \ldots (2) $
Adding $(1)$ and $ (2)$
$2 I=\frac{8}{\pi} \int_0^{\frac{\pi}{2}} \frac{(\cos x)^{2023}+(\sin x)^{2023}}{(\sin x)^{2023}+(\cos x)^{2023}} d x $
$2 I=\frac{8}{\pi} \int_0^{\frac{\pi}{2}} 1 dx$
$I=2$
