Answer:
Option C is correct.
Step-by-step explanation:
Your solution is correct. Given the integral [tex]\int \csc \left(4x\right)dx[/tex], let y = 4x, du = 4dx, respectively [tex]\frac{1}{4}[/tex]du = dx. Rewrite the given using these values of u and du.
[tex]\frac{du}{dx}[/tex] = 4 / 1 = 4,
[tex]\int csc(u)\frac{1}{4} du[/tex] - let us combine csc( u ) and 1 / 4,
[tex]\int csc(u) / 4 * du[/tex] - as 1 / 4 is a constant with respect to u, we can move it out of the integral,
[tex]\frac{1}{4} \int csc(u)du[/tex] - the integral of csc( u ) with respect to u is present in the form
" [tex]In(|csc(u) - cot(u)|)[/tex] . " Therefore,
[tex]\frac{1}{4} ( In(|csc(u) - cot(u)|) + C )[/tex] - replace all occurrences of u with 4x to receive the third solution. It can also be written as " [tex]\frac{1}{4} ( In(|csc(4x) - cot(4x)|) + C )[/tex], but only that your move the negative sign to the left of the integral.
