Question

What is the integral of $4{{x}^{3}}$?

Answer 1

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

& \Rightarrow I=4\left( \dfrac{{{x}^{4}}}{4} \right) \\\ & \Rightarrow I={{x}^{4}}.....(3) \\\ \end{aligned}$$ So, from equation (3) it is clear that the value of I is equal to $${{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.