[tex]\csc \left( x \right) = \frac{1}{{\sin \left( x \right)}} = \frac{{\sin \left( x\right)}}{{\sin {{\left( x \right)}^2}}} = \frac{{\sin \left( x\right)}}{{1 - {{\left( {\cos \left( x \right)} \right)}^2}}} \\ u = \cos \left( x \right){\rm{ }}\frac{{du}}{{dx}} = - \sin \left( x \right) \Rightarrow dx= - \frac{{du}}{{\sin \left( x \right)}} \\ \int {\csc \left( x \right)}{\rm{ }}dx = \int {\frac{1}{{\sin \left( x \right)}}} {\rm{ }}dx = \frac{{\sin \left( x \right)}}{{1 - {{\left( {\cos \left( x \right)}\right)}^2}}}dx = \int {\left( {\frac{{\sin \left( x \right)}}{{1 - {{\left( u\right)}^2}}}} \right)\left( { - \frac{{du}}{{\sin \left( x \right)}}}\right)} = - 1\int {\frac{1}{{1 - {u^2}}}du} = \\ - 1\int {\frac{1}{{2\left( {u + 1} \right)}} - \frac{1}{{2\left( {u - 1}\right)}}du} = \frac{1}{2}\left( {\ln \left( {1 - u} \right) - \ln \left( {u +1} \right)} \right) + C = \frac{1}{2}\ln \left( {\frac{{1 - u}}{{u + 1}}}\right) + C \\ \frac{{1 - \cos \left( x \right)}}{{\cos \left( x \right) + 1}}\left({\frac{{1 + \cos \left( x \right)}}{{1 + \cos \left( x \right)}}} \right) =\frac{{\sin {{\left( x \right)}^2}}}{{{{\left( {\cos \left( x \right) + 1}\right)}^2}}} = {\left( {\frac{{\sin \left( x \right)}}{{\left( {\cos \left( x\right) + 1} \right)}}} \right)^2} \Leftarrow kladd{\rm{ :p}} \\ \frac{1}{2}\ln \left( {\frac{{1 - u}}{{u + 1}}} \right) + C = \ln \left({\frac{{\sin \left( x \right)}}{{\left( {\cos \left( x \right) + 1} \right)}}}\right) + C \\ [/tex]
Sliter med omformingen fra der jeg er, svaret skal være
[tex]\ln(csc(x)-cot(x))+C[/tex] eller [tex]-\ln(csc(x)+cot(x))+C[/tex]
Noen som kan hjelpe ?
Klarte det
[tex]\int {\csc \left( x \right) = } \left\{ \begin{array}{l}\frac{1}{2}\ln \left( {\frac{{1 - \cos \left( x \right)}}{{1 + \cos \left( x\right)}}} \right) = \ln \left( {\sqrt {\frac{{1 - \cos \left( x \right)}}{{1 +\cos \left( x \right)}}} } \right) + C = \ln \left( {\tan \left( {\frac{x}{2}}\right)} \right) + C \\ \frac{1}{2}\ln \left( {\frac{{1 - \cos \left( x\right)}}{{1 + \cos \left( x \right)}}} \right) = \frac{1}{2}\ln\left({\frac{{1 - \cos \left( x \right)}}{{1 + \cos \left( x \right)}}\cdot\frac{{1 - \cos \left( x \right)}}{{1 - \cos \left( x \right)}}} \right) =\frac{1}{2}\ln \left( {\frac{{1 - \cos {{\left( x \right)}^2}}}{{\sin{{\left( x \right)}^2}}}} \right) = \ln \left( {\frac{{1 - \cos \left( x\right)}}{{\sin \left( x \right)}}} \right) = \ln \left( {\csc \left( x \right) -\cot \left( x \right)} \right) + C \\ \end{array} \right.\\\underline{\underline {{\rm{ }}\int {\csc \left( x \right) = } \left\{ \begin{array}{l}\ln \left( {\tan \left( {\frac{x}{2}} \right)} \right) + C \\ \ln \left( {\csc \left( x \right) - \cot \left( x \right)} \right) + C \\ \end{array} \right.{\rm{ }}}} \\ [/tex]
