Toán 12 Tích phân đặc biệt

tieutukeke said:

[tex]t=\frac{\pi }{2}-x\Rightarrow dx=-dt[/tex]
[tex]\Rightarrow I=\int_{\frac{\pi }{2}}^{0}\frac{sin^{n}t}{sin^{n}t+cos^{n}t}.(-dt)=\int_{0}^{\frac{\pi }{2}}\frac{sin^{n}t}{sin^{n}t+cos^{n}t}.dt=\int_{0}^{\frac{\pi }{2}}\frac{sin^{n}x}{sin^{n}x+cos^{n}x}.dx[/tex]
[tex]\Rightarrow I+I=\int_{0}^{\frac{\pi }{2}}\frac{cos^{n}x}{sin^{n}x+cos^{n}x}.dx+\int_{0}^{\frac{\pi }{2}}\frac{sin^{n}x}{sin^{n}x+cos^{n}x}.dx=\int_{0}^{\frac{\pi }{2}}dx=\frac{\pi }{2}\Rightarrow I=\frac{\pi }{4}[/tex]

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tieutukeke said:

[tex]t=\frac{\pi }{2}-x\Rightarrow dx=-dt[/tex]
[tex]\Rightarrow I=\int_{\frac{\pi }{2}}^{0}\frac{sin^{n}t}{sin^{n}t+cos^{n}t}.(-dt)=\int_{0}^{\frac{\pi }{2}}\frac{sin^{n}t}{sin^{n}t+cos^{n}t}.dt=\int_{0}^{\frac{\pi }{2}}\frac{sin^{n}x}{sin^{n}x+cos^{n}x}.dx[/tex]
[tex]\Rightarrow I+I=\int_{0}^{\frac{\pi }{2}}\frac{cos^{n}x}{sin^{n}x+cos^{n}x}.dx+\int_{0}^{\frac{\pi }{2}}\frac{sin^{n}x}{sin^{n}x+cos^{n}x}.dx=\int_{0}^{\frac{\pi }{2}}dx=\frac{\pi }{2}\Rightarrow I=\frac{\pi }{4}[/tex]

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