求不定积分(x^2-1)^(1/2) /x^2 dx
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![求不定积分(x^2-1)^(1/2) /x^2 dx](/uploads/image/z/2481427-19-7.jpg?t=%E6%B1%82%E4%B8%8D%E5%AE%9A%E7%A7%AF%E5%88%86%28x%5E2-1%29%5E%281%2F2%29+%2Fx%5E2+dx)
求不定积分(x^2-1)^(1/2) /x^2 dx
求不定积分(x^2-1)^(1/2) /x^2 dx
求不定积分(x^2-1)^(1/2) /x^2 dx
楼上怎么起了个头而没有尾啦?
令x=secN,dx=secNtanNdN
cosN=1/x,sinN=√(1-1/x²)=√(x²-1)/x
√(x²-1)=√(sec²N-1)=tanN
∴∫√(x²-1)/x² dx
=∫tanN/sec²N*secNtanN dN
=∫tan²N/secN dN
=∫sin²N/cosN dN
=∫(1-cos²N)/cosN dN
=∫(secN-cosN) dN
=ln|secN+tanN|-sinN+C,代入上面所设的换元
=ln|x+√(x²-1)|-√(x²-1)/x+C
令x=sect
原式=∫tant/(sect)^2d(sect)=∫tant^2/sectdt
=∫sectdt-∫sect^(-1)dt
=∫sectdt-∫costdt
= ln│sect+tant│+sint+C