∫(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
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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](/uploads/image/z/13791107-11-7.jpg?t=%E2%88%AB%28sinx%29%5E2%2F%28cosx%29%5E3dx+%E6%83%B3%E4%BA%86%E5%BE%88%E4%B9%85%E9%83%BD%E6%B2%A1%E5%81%9A%E5%87%BA%E6%9D%A5.+%E6%B1%82%E8%A7%A3%E5%95%8A%21%E7%AD%94%E6%A1%88%E5%A5%BD%E5%83%8F%E6%98%AF+++2%2Asinx%2F%28cosx%29%5E2%2Bln%28secx%2Btanx%29%2Bc)
∫(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
∫(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
∫(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
∫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