求∫sec³xdx,
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求∫sec³xdx,
![求∫sec³xdx,](/uploads/image/z/19788156-36-6.jpg?t=%E6%B1%82%E2%88%ABsec%26%23179%3Bxdx%2C)
原式=∫dx/cos³x
=∫cosxdx/(cos²x)²
=∫d(sinx)/(1-sin²x)²
=(1/4)∫[1/(1+sinx)+1/(1-sinx)+1/(1+sinx)²+1/(1-sinx)²]d(sinx)
=(1/4)[ln│1+sinx│-ln│1-sinx│-1/(1+sinx)+1/(1-sinx)]+C (C是积分常数)
=(1/4)[ln│(1+sinx)/(1-sinx)│+2sinx/(1-sin²x)]+C
=(1/4)[ln│(1+sinx)²/(1-sin²x)│+2sinx/(1-sin²x)]+C
=(1/4)[2ln│(1+sinx)/cosx│+2sinx/cos²x]+C
=(1/2)[ln│(1+sinx)/cosx│+sinx/cos²x]+C
=(1/2)(ln│secx+tanx│+secxtanx)+C.
=∫cosxdx/(cos²x)²
=∫d(sinx)/(1-sin²x)²
=(1/4)∫[1/(1+sinx)+1/(1-sinx)+1/(1+sinx)²+1/(1-sinx)²]d(sinx)
=(1/4)[ln│1+sinx│-ln│1-sinx│-1/(1+sinx)+1/(1-sinx)]+C (C是积分常数)
=(1/4)[ln│(1+sinx)/(1-sinx)│+2sinx/(1-sin²x)]+C
=(1/4)[ln│(1+sinx)²/(1-sin²x)│+2sinx/(1-sin²x)]+C
=(1/4)[2ln│(1+sinx)/cosx│+2sinx/cos²x]+C
=(1/2)[ln│(1+sinx)/cosx│+sinx/cos²x]+C
=(1/2)(ln│secx+tanx│+secxtanx)+C.