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

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

Explanation

Solution

Hint: In these types of questions always remember the basic integral values like ∫cos⁡ec2xdx=−cot⁡x  and  ∫cot⁡xcos⁡ecxdx=−cos⁡ecx\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 ∫cos⁡ecx(cos⁡ecx+cot⁡x)dx\int {\cos ecx(\cos ecx + \cot x)dx} be I.
So, I =∫cos⁡ecx(cos⁡ecx+cot⁡x)dx\int {\cos ecx(\cos ecx + \cot x)dx} (equation 1)
Now, on simplifying equation 1, we get
I = ∫(cos⁡ec2x+cot⁡xcos⁡ecx)dx\int {(\cos e{c^2}x} + \cot x\cos ecx)dx
It can also be written as,
I = ∫cos⁡ec2xdx+∫cot⁡xcos⁡ecxdx\int {\cos e{c^2}xdx} + \int {\cot x\cos ecxdx} (equation 2)
We know,
∫cos⁡ec2xdx=−cot⁡x&∫cot⁡xcos⁡ecxdx=−cos⁡ecx\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 −cosecx + c - cotx{\text{ }} - cosecx{\text{ }} + {\text{ }}c
I= −(cotx + cosecx)+c - \left( {cotx{\text{ }} + {\text{ }}cosecx} \right) + c
Hence ∫cos⁡ecx(cos⁡ecx+cot⁡x)dx\int {\cos ecx(\cos ecx + \cot x)dx} = −(cotx + cosecx)+c - \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.