Test your Integration Skills 102!

Calculus Level 3

$\large \int (2 + \sin^2 x) \cot x \, dx = \, ?$

Clarification : $C$ denotes the arbitrary constant of integration .

2 \ln (\sin (x) ) - (\cos^2 x) + C

2 \ln (\sin x) + \frac12 (\sin^2 x) + C

2 \csc (x) + \sin(x) \cos(x) + C

2 \csc(x) - (\sec^2 x) + C

1 solution

Kirubel Solomon
Mar 16, 2016

Step 1: Break down cot (x) into its constituents

       ∫ (2+ (sin (x)^2) cos(x)/sin(x)  dx

Step 2: Distribute then multiply

       ∫ (2 cos(x)/sin(x) + sin (x) cos (x)  dx

Step 3: Since its addition, you can break the integral into two parts

       ∫ (2 cos(x)/sin(x)   dx + ∫ sin (x) cos (x)   dx

Step 4: Now let u = sin (x) . Derivate u : dx = du/cos(x) . After this, replace u and cancel.

       ∫ 2/u  du + ∫ u  du  =  2 ln |u|+  u^2/2 + c

      2 ln |sin (x)|+ ((sin (x)^2)/2 + c