Please help me to evaluate this integral, without using csc function, because we don't use it on class, so must be some easier way to do it.
$\int{}\frac{\sqrt[7]{\operatorname{ctg}^3(x)}}{1-\cos^2 x}\mathrm dx$
Please help me to evaluate this integral, without using csc function, because we don't use it on class, so must be some easier way to do it.
$\int{}\frac{\sqrt[7]{\operatorname{ctg}^3(x)}}{1-\cos^2 x}\mathrm dx$
You mean $\displaystyle \int \frac{\cot^{3/7}(x)}{1-\cos^2(x)}\ dx$? Consider the substitution $u = \cot(x),\ du = \frac{-dx}{\sin^2(x)}$. Then $\displaystyle \int \frac{\cot^{3/7}(x)}{1-\cos^2(x)}\ dx = \int \frac{\cot^{3/7}(x)}{\sin^2(x)}\ dx = \int -u^{3/7}\, du = \frac{-7}{10} \cot^{10/7}(x) + C$.
Also, you can't really do this without (at least implicitly) using the $\csc(x)$ function. The cosecant function is defined as $\csc(x) = \frac{1}{\sin(x)}$, so as long as you have the definition of $\sin(x)$, you also have the definition of $\csc(x)$.