Question

Solve $\int {\cos ecx(\cos ecx + \cot x)dx} $.

Answer 1

Hint: In these types of questions always remember the basic integral values like $\int {\cos e{c^2}xdx = - \cot x \;and\; \int {\cot x \cos ecxdx} = - \cos ecx}$. Use these formulas to find the required simplifications.

Complete step-by-step answer:
Let $\int {\cos ecx(\cos ecx + \cot x)dx} $ be I.
So, I =$\int {\cos ecx(\cos ecx + \cot x)dx} $ (equation 1)
Now, on simplifying equation 1, we get
I = \[\int {(\cos e{c^2}x} + \cot x\cos ecx)dx\]
It can also be written as,
I = $\int {\cos e{c^2}xdx} + \int {\cot x\cos ecxdx} $ (equation 2)
We know,
\[\int {\cos e{c^2}xdx = - \cot x\& \int {\cot x\cos ecxdx} = - \cos ecx} \] (equation 3)
Substituting values of equation 3 in equation 2 gives us,
I = \[ - cotx{\text{ }} - cosecx{\text{ }} + {\text{ }}c\]
I= \[ - \left( {cotx{\text{ }} + {\text{ }}cosecx} \right) + c\]
Hence $\int {\cos ecx(\cos ecx + \cot x)dx} $= \[ - \left( {cotx{\text{ }} + {\text{ }}cosecx} \right) + c\].

Note: Try to memorize as many formulas as possible because it will give you a boost to solve questions and save your time. Always simplify the question by dividing it into familiar form and then substitute it with the simplified value.