Question

How do you integrate $\left( {\dfrac{x}{{x + 2}}} \right)$ ?

Answer 1

Hint:In this question, first we have to add and subtract $2$ in the numerator. Then we have to split the numerator in two terms. In the first term numerator and denominator will have the same part and the second term will also be in the integrable format. Then we have to do just simple integration.

Complete step by step answer:
In the above question, we have to do integration of the given term
$ \Rightarrow \int {\dfrac{x}{{x + 2}}dx} $
Now, add and subtract $2$ in the numerator.
$ \Rightarrow \int {\dfrac{{x + 2 - 2}}{{x + 2}}dx} $
$ \Rightarrow \int {\dfrac{{x + 2}}{{x + 2}}dx} - \int {\dfrac{2}{{x + 2}}dx} $
Now, on cancelling the numerator and denominator part in the first term.
$ \Rightarrow \int {dx - 2\int {\dfrac{1}{{x + 2}}dx} } $
Now we know that $\int {dx = x} $ and $\int {\dfrac{1}{x}dx} = \ln x$. Similarly, $\int {\dfrac{1}{{1 + x}}dx} = \ln \left( {1 + x} \right)$
$ \Rightarrow x - 2\ln \left( {x + 2} \right) + C$
Here, C is the constant of integration.
Therefore, the value of required integral $x - 2\ln \left( {x + 2} \right) + C$.

Note:We can also do this question by substituting x with trigonometric functions line sine function or cosine function. But the above method is the simplest one. We can also check our answer if it is correct or not by differentiating it. If we get the same result as the given question, then we have done it correctly.