Question

Question 8
Find the indefinite integral.
csc(4x) dx.

Answer 1

Option C is correct.

Explanation:

Your solution is correct. Given the integral
`\int \csc \left(4x\right)dx`, let y = 4x, du = 4dx, respectively
`(1)/(4)`du = dx. Rewrite the given using these values of u and du.

`(du)/(dx)` = 4 / 1 = 4,

`\int csc(u)(1)/(4) du` - let us combine csc( u ) and 1 / 4,

`\int csc(u) / 4 * du` - as 1 / 4 is a constant with respect to u, we can move it out of the integral,

`(1)/(4) \int csc(u)du` - the integral of csc( u ) with respect to u is present in the form

"
`In(|csc(u) - cot(u)|)` . " Therefore,

`(1)/(4) ( In(|csc(u) - cot(u)|) + C )` - replace all occurrences of u with 4x to receive the third solution. It can also be written as "
`(1)/(4) ( In(|csc(4x) - cot(4x)|) + C )`, but only that your move the negative sign to the left of the integral.

Answer 2

The correct answer is C. Using u-substitution: u=4x, du/dx=4, dx=du/4. So now we have to take the indefinite integral of (1/4)*csc(u). Which is (-1/4)ln|csc(u)+cot(u)|. Finally plug in 4x for u, which gives C, or the third option.