Question

Substitution (SW Question 13, Use a change of variables or the table to evaluate the following indefinite integral. csc? dx cotx Click the icon to view the table of general integration formulas. csc?x dx= col X e the follo Integration Formulas cos ax dx sin ax+C sin ax dx = cos ax + C a Integration fo sec?ax dx = tan ax + CSC ax dx- cot ax+C sec axtan ax dx = sec ax + c a csc ax cotax dx- CSC ax + C [ Sescax 16*dx = 160* +0,620, 641 S -- sin.c.a> o 3dx +C Inb dx dx tan 2.C a +x dx مد و مداء 11 - Print Done Clear all

Answer 1

Answer: Note that we used the table of general integration formulas to recognize that the integral of csc(x) dx is - ln|csc(x) + cot(x)| + C.

Explanation:

To evaluate the indefinite integral of csc(x) cot(x) dx, we can use substitution.

Let u = cot(x), then du/dx = -csc^2(x) and dx = -du/csc^2(x).

Substituting these values in the integral, we get:

∫ csc(x) cot(x) dx = ∫ -du/u = -ln|u| + C

Now substituting back u = cot(x),

we get: ∫ csc(x) cot(x) dx = -ln|cot(x)| + C

This is the final answer.

Note that we used the table of general integration formulas to recognize that the integral of csc(x) dx is -ln|csc(x) + cot(x)| + C. We then used the substitution technique and the general integration formula for ln|u| to arrive at the final answer.