Question

What is the anti – derivative of $f\left( x \right)=5$?

Answer 1

Assume the integral of the given function as ‘I’. Now, integrate the given function with respect to $dx$ given as $I=\int{f\left( x \right)dx}$. Use the basic integral formula $\int{{{x}^{n}}dx}=\dfrac{{{x}^{n+1}}}{n+1}$ for $n\ne -1$ to integrate the function. To use this formula for the constant term 5, write it as $5{{x}^{0}}$ and then evaluate. Add the constant of indefinite integration ‘C’ at last to get the answer.

Complete step-by-step solution:
Here we have been provided with the function $f\left( x \right)=5$ and we are asked to find its anti – derivative. In other words we need to integrate this function. Let us assume the integral as I, so we have,
$\Rightarrow I=\int{5dx}$
Now, since 5 is a constant so we can write it as $5\times 1$. Further 1 can be written as the function of x with exponent equal to 0 as ${{x}^{0}}$, so we have,
$\Rightarrow I=\int{5{{x}^{0}}dx}$
Here 5 is a constant so it can be taken out of the integral. So we get,
$\Rightarrow I=5\int{{{x}^{0}}dx}$
Now, applying the basic formula of integral given as: - $\int{{{x}^{n}}dx}=\dfrac{{{x}^{n+1}}}{n+1}$, we get,

& \Rightarrow I=5\left( \dfrac{{{x}^{0+1}}}{0+1} \right) \\\ & \Rightarrow I=5x \\\ \end{aligned}$$ Now, since the given integral is an indefinite integral and therefore we need to add a constant of integration (C) in the expression obtained for I. So we get, $$\therefore I=5x+C$$ Hence, the above relation is our answer. **Note:** One may note that the basic formula that we have used to find the integral I given as: - $$\int{{{x}^{n}}dx}=\dfrac{{{x}^{n+1}}}{n+1}$$ is invalid for n = -1. This is because in this case (n + 1) will become 0 and the integral will become undefined. So, when n = -1 then the function becomes $$\int{\dfrac{1}{x}dx}$$ whose solution is $$\ln x$$. You must remember all the basic formulas of indefinite integrals and that for the different functions. At last, do not forget to add the constant of integration (C) as we are finding indefinite integral and not definite integral.