Question

What is the integral of \[4{{x}^{3}}\]?

Answer 1

Hint: We know that \[\int{cf(x)dx=c\int{f(x)dx}}\] where c be any constant. We know that the integral of \[{{x}^{n}}\] is equal to \[\dfrac{{{x}^{n+1}}}{n+1}\]. By using these integration concepts and formulae, this problem can be solved in order to correct answers.

Complete step-by-step answer:
From the question, it is clear that we have to find the integral of \[4{{x}^{3}}\].
Let us assume that the value of integral of \[4{{x}^{3}}\] is equal to I.
\[\Rightarrow I=\int{4{{x}^{3}}}dx\]
Let us assume this as equation (1),
\[\Rightarrow I=\int{4{{x}^{3}}}dx...(1)\]
We know that \[\int{cf(x)dx=c\int{f(x)dx}}\] where c be any constant.
Now we will apply this concept in equation (1), then we get
\[\Rightarrow I=4\int{{{x}^{3}}}dx\]
Let us assume this as equation (2), then we get
\[\Rightarrow I=4\int{{{x}^{3}}}dx.....(2)\]
We know that the integral of \[{{x}^{n}}\] is equal to \[\dfrac{{{x}^{n+1}}}{n+1}\].
Now we will apply this concept in equation (3).
First let us compare \[{{x}^{n}}\] with \[{{x}^{3}}\].
Now it is clear that the value of n is equal to 3.
Now from equation (3), we get
\[\Rightarrow I=4\left( \dfrac{{{x}^{3+1}}}{3+1} \right)\]
Now by simplification, we get
\[\begin{align}
  & \Rightarrow I=4\left( \dfrac{{{x}^{4}}}{4} \right) \\
 & \Rightarrow I={{x}^{4}}.....(3) \\
\end{align}\]
So, from equation (3) it is clear that the value of I is equal to \[{{x}^{4}}\].
So, finally we can conclude that the integral of \[4{{x}^{3}}\] is equal to \[{{x}^{4}}\].

Note: Students may have a misconception that the integral of \[{{x}^{n}}\] is equal to \[\dfrac{{{x}^{n+1}}}{n}\] but we know that the integral of \[{{x}^{n}}\] is equal to \[\dfrac{{{x}^{n+1}}}{n+1}\]. If this misconception is followed, then the final answer may get interrupted. Students should also be aware such that no calculation mistake is done while solving the problem because if a small mistake is done then the final answer will get interrupted.