Question

What is the value of integral $\int{\cot x\cos xdx}?$

Answer 1

Hint: We have already learnt the trigonometric identity given by $\cot x=\dfrac{\cos x}{\sin x}.$ And we also know the trigonometric identity given by ${{\sin }^{2}}x+{{\cos }^{2}}x=1.$ We will rearrange the identities if necessary. Then we will use the linearity property of integration.

Complete step-by-step solution:
Let us consider the given integral $\int{\cot x\cos xdx}.$
In order to find the integral, we need to simplify the function under the integral side.
So, we will use the trigonometric identity given by $\cot x=\dfrac{\cos x}{\sin x}.$
Let us substitute this in the given integral.
Then, we will get $\int{\cot x\cos xdx}=\int{\dfrac{\cos x}{\sin x}\cos xdx}.$
Now, we can change this into $\int{\cot x\cos xdx}=\int{\dfrac{{{\cos }^{2}}x}{\sin x}dx}.$
We have already learnt the Pythagorean identity given by ${{\sin }^{2}}x+{{\cos }^{2}}x=1.$
Now, if we rearrange this trigonometric identity, we will get ${{\cos }^{2}}x=1-{{\sin }^{2}}x.$
Let us substitute this identity in the given integral.
Then, we will get $\int{\cot x\cos xdx}=\int{\dfrac{1-{{\sin }^{2}}x}{\sin x}dx}.$
From this, we will get $\int{\cot x\cos xdx}=\int{\left( \dfrac{1}{\sin x}-\dfrac{{{\sin }^{2}}x}{\sin x} \right)dx}.$
Now let us use the linearity property of the integrals.
We know that the linear property is given by $\int{\left( f+g \right)dx}=\int{fdx}+\int{gdx}$ where $f$ and $g$ are functions of $x.$
Now, when we apply this linearity property, we will get $\int{\cot x\cos xdx}=\int{\dfrac{1}{\sin x}dx}-\int{\dfrac{{{\sin }^{2}}x}{\sin x}dx}.$
We know that $\dfrac{1}{\sin x}=\cos ecx$ and $\dfrac{{{\sin }^{2}}x}{\sin x}=\sin x.$
Now, when we substitute these identities, we will get $\int{\cot x\cos xdx}=\int{\cos ecxdx}-\int{\sin xdx}.$
Let us recall the integrals given by $\int{\sin xdx}=-\cos x+C$ and $\int{\cos ecx}dx=-\ln \left| \cos ecx+\cot x \right|+C$
Now, we are going to apply the above written identities in the obtained integral.
Then, as a result, we will get $\int{\cot x\cos xdx}=-\ln \left| \cos ecx+\cot x \right|-\left( -\cos x \right)+C$ where $C$ is the constant of integration.
And this will give us $\int{\cot x\cos xdx}=-\ln \left| \cos ecx+\cot x \right|+\cos x+C.$
Hence the required integral is $\int{\cot x\cos xdx}=-\ln \left| \cos ecx+\cot x \right|+\cos x+C.$

Note: We should not forget to put the constant of integration when we deal with the indefinite integrals where there are no upper limit and lower limit given. The integrals with upper limit and lower limit are called the definite integrals.