∫(sinx)^2/(cosx)^3dx 想了很久都没做出来. 求解啊!
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∫(sinx)^2/(cosx)^3dx 想了很久都没做出来. 求解啊!
答案好像是 2*sinx/(cosx)^2+ln(secx+tanx)+c
答案好像是 2*sinx/(cosx)^2+ln(secx+tanx)+c
∫sin²x/cos³x dx
= ∫(sin²x/cos²x)(1/cosx) dx
= ∫tan²x*secx dx
= ∫(sec²x-1)*secx dx
= ∫sec³x dx - ∫secx dx
= I - J
J = ∫secx dx
= ∫secx*(secx+tanx)/(secx+tanx) dx
= ∫(secx*tanx+sec²x)/(secx+tanx)
= ∫d(secx+tanx)/(secx+tanx)
= ln|secx+tanx| + C
I = ∫sec³x dx
= ∫sec²x*secx dx
= ∫secx dtanx
= secx*tanx - ∫tanx d(secx)
= secx*tanx - ∫tan²x*secx dx
= secx*tanx - ∫(sec²x-1)*secx dx
= secx*tanx - ∫sec³x dx + ∫secx dx
= secx*tanx - I + J
2I = secx*tanx + J
I = (1/2)secx*tanx + (1/2)J
= (1/2)secx*tanx + (1/2)ln|secx+tanx|
原式= I - J
= (1/2)secx*tanx + (1/2)J - J
= (1/2)secx*tanx - (1/2)ln|secx+tanx| + C
= ∫(sin²x/cos²x)(1/cosx) dx
= ∫tan²x*secx dx
= ∫(sec²x-1)*secx dx
= ∫sec³x dx - ∫secx dx
= I - J
J = ∫secx dx
= ∫secx*(secx+tanx)/(secx+tanx) dx
= ∫(secx*tanx+sec²x)/(secx+tanx)
= ∫d(secx+tanx)/(secx+tanx)
= ln|secx+tanx| + C
I = ∫sec³x dx
= ∫sec²x*secx dx
= ∫secx dtanx
= secx*tanx - ∫tanx d(secx)
= secx*tanx - ∫tan²x*secx dx
= secx*tanx - ∫(sec²x-1)*secx dx
= secx*tanx - ∫sec³x dx + ∫secx dx
= secx*tanx - I + J
2I = secx*tanx + J
I = (1/2)secx*tanx + (1/2)J
= (1/2)secx*tanx + (1/2)ln|secx+tanx|
原式= I - J
= (1/2)secx*tanx + (1/2)J - J
= (1/2)secx*tanx - (1/2)ln|secx+tanx| + C
∫(sinx)^2/(cosx)^3dx 想了很久都没做出来. 求解啊!
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∫(sinx)^2/(cosx)^3 dx
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这个怎么求啊~∫ ((sinx)^2/(cosx)^3)dx
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请教一道数学题,做了很久没做出来
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