Question

How do you integrate $ \dfrac{x}{x+1}dx$?

Answer 1

Hint: To solve the given integral we will first subtract and add 1 from the numerator. Then we use the integration formula to solve further. Following the integration formula we will use in order to solve the question:
$\int{cdx=x}$ where c is any constant term.
$\int{\dfrac{1}{x}=\ln |x|}$

Complete step by step answer:
We have been given an expression $ \dfrac{x}{x+1}dx$
We have to integrate the given expression.
We know that integration is the inverse of differentiation. Integral of a function generally differ by numbers. An indefinite integral does not contain upper and lower limits. An indefinite integral is represented as
$\int{f(x)dx=f(x)+c}$ where, c is the arbitrary constant or integration constant.
We have $ \dfrac{x}{x+1}dx$
Now, we have to integrate the given function we get
 $ \Rightarrow \int{\dfrac{x}{x+1}dx}$
Now, subtract and add 1 from the numerator we get
$\Rightarrow \int{\dfrac{x+1-1}{x+1}dx}$
Now, we can rewrite the obtained equation as
$\Rightarrow \int{\dfrac{x+1}{x+1}dx}-\int{\dfrac{1}{x+1}dx}$
Now, simplifying the obtained equation we get
$\Rightarrow \int{1dx}-\int{\dfrac{1}{x+1}dx}$
Now, we know that $ \int{cdx=x}$ where, c is any constant term and $ \int{\dfrac{1}{x}=\ln |x|}$
Now, applying the above integration formula we get
$\Rightarrow x-\ln \left| x+1 \right|+c$
Hence on integrating $ \dfrac{x}{x+1}dx$ we get $ x-\ln \left| x+1 \right|+c$.

Note:
The indefinite integral is the primitive integral of a differentiable function whose derivative is equal to the original function. Here the given function is not directly integrated because we don’t have any formula to solve such type of function so we need to convert it into another form. So students must have the knowledge of basic integrals to solve the questions.