Question

Integrate 2xdx

Answer 1

Hint: To solve the question, we have to apply the formula of integration to isolate the integration function of the constant values and then apply the formula of integration for the given function of x.

Complete step-by-step answer:
Let the given expression for integration be equal to f(x).
 \[\Rightarrow \] f(x) = 2x dx
The integral of 2x dx is now equal to the integral of f(x).
\[\Rightarrow \int{f(x)=\int{2x}dx}\]
We know that the integral of function of a variable with constant coefficient is independent of the constant. The formula for function of a variable with constant coefficient is given by \[\int{af(x)=a\int{f(x)dx}}\]
Where a is a constant and f(x) is a function of variable x.
Thus, on applying the above formula to the given equation, we get
\[\int{f(x)=2\int{xdx}}\]
We know that the formula for integration of \[{{x}^{n}}\] is given by \[\int{{{x}^{n}}dx}=\dfrac{{{x}^{n+1}}}{n+1}+c\] where c is an integration constant.
 On comparing with our integration function, we get n = 1. On applying the above formula to the integration function, we get
\[\int{f(x)=2\left( \dfrac{{{x}^{1+1}}}{1+1} \right)+c}\]
\[\int{f(x)=2\left( \dfrac{{{x}^{2}}}{2} \right)+c}\]
\[\int{f(x)={{x}^{2}}+c}\]
Thus, the value of integration of 2x dx is equal to \[{{x}^{2}}+c\]
Note: The possibility of mistake can be not applying appropriate formula for integration of the given function of x. The other possibility of mistake can be not analysing that integration can be done only to variable function thus being independent of the constant coefficient of the given expression. The possibility of landing with the wrong answer can be removed since we can check the answer by differentiating the obtained answer. The formula for differentiation of \[{{x}^{n}}\] is equal to \[n{{x}^{n-1}}\]. Thus, we can check our answer by substituting n = 2 in the mentioned formula.