Question

can anyone show me how cscx=ln|cscx-cotx|+C

Answer 1

\[\int\limits_{}^{}cscx \space dx\]

multiply top and bottom my cscx + cotx

\[\int\limits_{}^{}cscx \frac{cscx + cotx}{cscx + cotx} \space dx\]

let u= cscx + cotx and du = (-cscxcotx - csc^2x)dx

\[- \int\limits_{}^{}\frac{(-cscxcotx - \csc^2x)}{cscx + cotx} \space dx\] now do the substitution
\[-\int\limits_{}^{} \frac{1}{u}\space du\]

\[-\ln|u| + C\] replace u with the above expression

\[-\ln|cscx + cotx| + C\]