The value of 8_0^/2( x)^2023( x)^2023+( x)^2023,dx is ____ Answer: 2

Question

The value of $\dfrac{8}{\pi}\displaystyle\int_0^{\pi/2}\frac{(\cos x)^{2023}}{(\sin x)^{2023}+(\cos x)^{2023}}\,dx$ is \_\_\_\_

Answer: 2

Accepted Answer

Let $I = \dfrac8\pi\displaystyle\int_0^{\pi/2}\dfrac{(\cos x)^{2023}}{(\sin x)^{2023}+(\cos x)^{2023}}\,dx$. Using $\int_0^af(x)dx=\int_0^af(a-x)dx$:

$$I = \frac8\pi\int_0^{\pi/2}\frac{(\sin x)^{2023}}{(\sin x)^{2023}+(\cos x)^{2023}}\,dx$$

Adding both expressions for $I$:

$$2I = \frac8\pi\int_0^{\pi/2}1\,dx = \frac8\pi\cdot\frac\pi2 = 4 \implies I=2$$

Solutions · 1