Question

Find $\int{\cos ecx\left( \cos ecx+\cot x \right)dx}$?

Answer 1

In the given question, we have been asked to integrate the function. In order to solve the questions, we have to use trigonometric integration formulas to simplify the given integration function and then follow integration formulas or methods to integrate. We have to split the given trigonometric terms by using identities. Perform further integration in the simplified form and hence we get the required solution.

Complete step-by-step solution:
We have given that,
$\int{\cos ecx\left( \cos ecx+\cot x \right)dx}$
Let I be the integral, such that
$\Rightarrow I=\int{\cos ecx\left( \cos ecx+\cot x \right)dx}$
Simplifying the above given integral, we will obtain
$\Rightarrow I=\int{\left( \cos e{{c}^{2}}x+\cos ecx\cot x \right)dx}$
Separating the integral into the sum of two terms,
We will get
$\Rightarrow I=\int{\cos e{{c}^{2}}xdx}+\int{\cos ecx\cot dx}$
Applying the trigonometric integration properties,
We know that,
$\int{\cos e{{c}^{2}}xdx}=-\cot x+C$ and $\int{\cos ecx\cot dx}=-\cos ecx+C$
Therefore,
Combining the two, we will get
$\Rightarrow I=\int{\cos e{{c}^{2}}xdx}+\int{\cos ecx\cot dx}=-\cot x-\cos ecx+C$
Thus,
$\int{\cos ecx\left( \cos ecx+\cot x \right)dx}=-\cot x-\cos ecx+C$
Hence, this is the required integration.

Note: When doing indefinite integration, always write +C part after the integration. This +C part indicates the constant part remains after integration and can be understood when you explore it graphically. The finite integration constant gets cancelled out, so we only write it in indefinite integration. To solve any given numerical, or function, there are different types of integration methods like integration by substitution, integration by parts, etc. We should remember the property or the formulas of integration, this would make it easier to solve the question. You should always remember all the methods for integration so that we can easily choose which method is suitable for solving the particular type of question. We should do all the calculations carefully and explicitly to avoid making errors.