Respuestas
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1
recuerda que
entonces hacemos lo siguiente
![\displaystyle
\int \csc^3 x\, dx=-\int \csc x\cdot (-\csc^2x)\, dx\\ \\ \\
\int \csc^3 x\, dx=-\int \csc x\, d(\cot x)\\ \\ \\
\int \csc^3 x\, dx=-\csc x\cdot\cot x+\int \cot x\cdot d(\csc x)\\ \\ \\
\int \csc^3 x\, dx=-\csc x\cdot\cot x+\int \cot x (-\csc x \cot x)\,dx\\ \\ \\
\int \csc^3 x\, dx=-\csc x\cdot\cot x-\int\csc x\cot^2 x\, dx
\displaystyle
\int \csc^3 x\, dx=-\int \csc x\cdot (-\csc^2x)\, dx\\ \\ \\
\int \csc^3 x\, dx=-\int \csc x\, d(\cot x)\\ \\ \\
\int \csc^3 x\, dx=-\csc x\cdot\cot x+\int \cot x\cdot d(\csc x)\\ \\ \\
\int \csc^3 x\, dx=-\csc x\cdot\cot x+\int \cot x (-\csc x \cot x)\,dx\\ \\ \\
\int \csc^3 x\, dx=-\csc x\cdot\cot x-\int\csc x\cot^2 x\, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%0A%5Cint+%5Ccsc%5E3+x%5C%2C+dx%3D-%5Cint+%5Ccsc+x%5Ccdot+%28-%5Ccsc%5E2x%29%5C%2C+dx%5C%5C+%5C%5C+%5C%5C%0A%5Cint+%5Ccsc%5E3+x%5C%2C+dx%3D-%5Cint+%5Ccsc+x%5C%2C+d%28%5Ccot+x%29%5C%5C+%5C%5C+%5C%5C%0A%5Cint+%5Ccsc%5E3+x%5C%2C+dx%3D-%5Ccsc+x%5Ccdot%5Ccot+x%2B%5Cint+%5Ccot+x%5Ccdot+d%28%5Ccsc+x%29%5C%5C+%5C%5C+%5C%5C%0A%5Cint+%5Ccsc%5E3+x%5C%2C+dx%3D-%5Ccsc+x%5Ccdot%5Ccot+x%2B%5Cint+%5Ccot+x+%28-%5Ccsc+x+%5Ccot+x%29%5C%2Cdx%5C%5C+%5C%5C+%5C%5C%0A%5Cint+%5Ccsc%5E3+x%5C%2C+dx%3D-%5Ccsc+x%5Ccdot%5Ccot+x-%5Cint%5Ccsc+x%5Ccot%5E2+x%5C%2C+dx%0A%0A%0A)
![\displaystyle
\int \csc^3 x\, dx=-\csc x\cdot\cot x-\int\csc x\cdot (\csc^2x-1)\, dx\\ \\ \\
\int \csc^3 x\, dx=-\csc x\cdot\cot x-\int \csc^3x-\csc x \, dx\\ \\ \\
2\int \csc^3 x\, dx=-\csc x\cdot\cot x+\int \csc x \, dx\\ \\ \\
2\int \csc^3 x\, dx=-\csc x\cdot\cot x+\int \dfrac{1}{\sin x} \, dx\\ \\ \\
2\int \csc^3 x\, dx=-\csc x\cdot\cot x+\int \dfrac{\sin x}{\sin^2 x} \, dx \displaystyle
\int \csc^3 x\, dx=-\csc x\cdot\cot x-\int\csc x\cdot (\csc^2x-1)\, dx\\ \\ \\
\int \csc^3 x\, dx=-\csc x\cdot\cot x-\int \csc^3x-\csc x \, dx\\ \\ \\
2\int \csc^3 x\, dx=-\csc x\cdot\cot x+\int \csc x \, dx\\ \\ \\
2\int \csc^3 x\, dx=-\csc x\cdot\cot x+\int \dfrac{1}{\sin x} \, dx\\ \\ \\
2\int \csc^3 x\, dx=-\csc x\cdot\cot x+\int \dfrac{\sin x}{\sin^2 x} \, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%0A%5Cint+%5Ccsc%5E3+x%5C%2C+dx%3D-%5Ccsc+x%5Ccdot%5Ccot+x-%5Cint%5Ccsc+x%5Ccdot+%28%5Ccsc%5E2x-1%29%5C%2C+dx%5C%5C+%5C%5C+%5C%5C%0A%5Cint+%5Ccsc%5E3+x%5C%2C+dx%3D-%5Ccsc+x%5Ccdot%5Ccot+x-%5Cint+%5Ccsc%5E3x-%5Ccsc+x+%5C%2C+dx%5C%5C+%5C%5C+%5C%5C%0A2%5Cint+%5Ccsc%5E3+x%5C%2C+dx%3D-%5Ccsc+x%5Ccdot%5Ccot+x%2B%5Cint+%5Ccsc+x+%5C%2C+dx%5C%5C+%5C%5C+%5C%5C%0A2%5Cint+%5Ccsc%5E3+x%5C%2C+dx%3D-%5Ccsc+x%5Ccdot%5Ccot+x%2B%5Cint+%5Cdfrac%7B1%7D%7B%5Csin+x%7D+%5C%2C+dx%5C%5C+%5C%5C+%5C%5C%0A2%5Cint+%5Ccsc%5E3+x%5C%2C+dx%3D-%5Ccsc+x%5Ccdot%5Ccot+x%2B%5Cint+%5Cdfrac%7B%5Csin+x%7D%7B%5Csin%5E2+x%7D+%5C%2C+dx)
![\displaystyle
2\int \csc^3 x\, dx=-\csc x\cdot\cot x+\int \dfrac{\sin x}{1-\cos^2 x} \, dx\\ \\ \\
2\int \csc^3 x\, dx=-\csc x\cdot\cot x-\int \dfrac{d(\cos x)}{1-\cos^2 x} \\ \\ \\
\text{Recuerde que... } \int \dfrac{1}{1-u^2}\, du=\dfrac{1}{2}\ln \left|\dfrac{1+u}{1-u}\right|+C\\ \\ \\
2\int \csc^3 x\, dx=-\csc x\cdot\cot x-\dfrac{1}{2}\ln \left|\dfrac{1+\cos x}{1-\cos x}\right|+C
\displaystyle
2\int \csc^3 x\, dx=-\csc x\cdot\cot x+\int \dfrac{\sin x}{1-\cos^2 x} \, dx\\ \\ \\
2\int \csc^3 x\, dx=-\csc x\cdot\cot x-\int \dfrac{d(\cos x)}{1-\cos^2 x} \\ \\ \\
\text{Recuerde que... } \int \dfrac{1}{1-u^2}\, du=\dfrac{1}{2}\ln \left|\dfrac{1+u}{1-u}\right|+C\\ \\ \\
2\int \csc^3 x\, dx=-\csc x\cdot\cot x-\dfrac{1}{2}\ln \left|\dfrac{1+\cos x}{1-\cos x}\right|+C](https://tex.z-dn.net/?f=%5Cdisplaystyle%0A2%5Cint+%5Ccsc%5E3+x%5C%2C+dx%3D-%5Ccsc+x%5Ccdot%5Ccot+x%2B%5Cint+%5Cdfrac%7B%5Csin+x%7D%7B1-%5Ccos%5E2+x%7D+%5C%2C+dx%5C%5C+%5C%5C+%5C%5C%0A2%5Cint+%5Ccsc%5E3+x%5C%2C+dx%3D-%5Ccsc+x%5Ccdot%5Ccot+x-%5Cint+%5Cdfrac%7Bd%28%5Ccos+x%29%7D%7B1-%5Ccos%5E2+x%7D+%5C%5C+%5C%5C+%5C%5C%0A%5Ctext%7BRecuerde+que...+%7D++%5Cint+%5Cdfrac%7B1%7D%7B1-u%5E2%7D%5C%2C+du%3D%5Cdfrac%7B1%7D%7B2%7D%5Cln+%5Cleft%7C%5Cdfrac%7B1%2Bu%7D%7B1-u%7D%5Cright%7C%2BC%5C%5C+%5C%5C+%5C%5C%0A2%5Cint+%5Ccsc%5E3+x%5C%2C+dx%3D-%5Ccsc+x%5Ccdot%5Ccot+x-%5Cdfrac%7B1%7D%7B2%7D%5Cln+%5Cleft%7C%5Cdfrac%7B1%2B%5Ccos+x%7D%7B1-%5Ccos+x%7D%5Cright%7C%2BC%0A%0A)
![\displaystyle
2\int \csc^3 x\, dx=-\csc x\cdot\cot x-\ln |\csc x + \cot x|+C\\ \\ \\
\int \csc^3 x\, dx=-\dfrac{1}{2}\csc x\cdot\cot x-\dfrac{1}{2}\ln |\csc x + \cot x|+C\\ \\ \\ \displaystyle
2\int \csc^3 x\, dx=-\csc x\cdot\cot x-\ln |\csc x + \cot x|+C\\ \\ \\
\int \csc^3 x\, dx=-\dfrac{1}{2}\csc x\cdot\cot x-\dfrac{1}{2}\ln |\csc x + \cot x|+C\\ \\ \\](https://tex.z-dn.net/?f=%5Cdisplaystyle%0A2%5Cint+%5Ccsc%5E3+x%5C%2C+dx%3D-%5Ccsc+x%5Ccdot%5Ccot+x-%5Cln+%7C%5Ccsc+x+%2B+%5Ccot+x%7C%2BC%5C%5C+%5C%5C+%5C%5C%0A%5Cint+%5Ccsc%5E3+x%5C%2C+dx%3D-%5Cdfrac%7B1%7D%7B2%7D%5Ccsc+x%5Ccdot%5Ccot+x-%5Cdfrac%7B1%7D%7B2%7D%5Cln+%7C%5Ccsc+x+%2B+%5Ccot+x%7C%2BC%5C%5C+%5C%5C+%5C%5C)
Anónimo:
Muchas Gracias.. y me puedes ayudar con esta ∫sec³(x)dx
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