Tính tích phân sau: ∫sin^3 x /3^x+3^(−x)dx

Question

Tính tích phân sau:

\(\int _{-\frac{\pi }{2} }^{\frac{\pi }{2} }\frac{\sin ^{3} x}{3^{x} +3^{-x} } {\rm d}x .\)

Answer 1

\(I =\int _{\frac{-\pi }{2} }^{\frac{\pi }{2} }\ln (x+\sqrt{x^{2} +1} ) \, dx \) 

 Đặt  \(x=-t\Rightarrow dx=-dt,\) khi đó :
\(I =\int _{\frac{-\pi }{2} }^{\frac{\pi }{2} }\ln (x+\sqrt{x^{2} +1} ) \, dx\)

\(=-\int _{\frac{\pi }{2} }^{\frac{-\pi }{2} }\ln (-x+\sqrt{x^{2} +1} \, ) \, dx \)
\(=\int _{\frac{-\pi }{2} }^{\frac{\pi }{2} }\ln (-x+\sqrt{x^{2} +1} \, ) \, dx=\int _{\frac{-\pi }{2} }^{\frac{\pi }{2} }\ln \frac{1}{x+\sqrt{x^{2} +1} }  \, dx  \)

\(=-\int _{\frac{-\pi }{2} }^{\frac{\pi }{2} }\ln (x+\sqrt{x^{2} +1} ) \, dx=-I . \)
Suy ra :    \(2I =0\Rightarrow I =0\)

 Đáp số :  \( I =0\).