csc^2x dx
-cotx + c
Integrating csc^2x dx requires the use of integration by substitution.
Step 1: Substitute u = sin x. Then, du/dx = cos x dx, and dx = du/cos x.
Step 2: Replace dx in the integral with du/cos x.
csc^2 x dx = csc^2 x (du/cos x)
Step 3: Simplify by using the identity, csc^2 x = 1/sin^2 x.
csc^2 x (du/cos x) = (1/sin^2 x) (du/cos x)
Step 4: Substitute sin^2 x = 1 – cos^2 x.
(1/sin^2 x) (du/cos x) = (1/(1 – cos^2 x))(du/cos x)
Step 5: Multiply and divide by negative one to make the expression easier to integrate.
(1/(cos^2 x – 1))(-du/-cos x)
Step 6: Replace cos^2 x with (1 – sin^2 x) and simplify.
(1/(1 – sin^2 x – 1))(-du/-cos x)
(-1/cos x)(du/(sin^2 x)) = (-du/(cos x*sin^2 x))
The final result is -cot x + C.
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