Question

How do you find the antiderivative of $f\left( x \right)=8{{x}^{3}}+5{{x}^{2}}-9x+3$?

Answer 1

Assume the value of the given integral as ‘I’. Break the integral into four parts and find the integral of each of the terms: - $8{{x}^{3}},5{{x}^{2}},-9x$ and 3. Use the basic integral formula, $\int{{{x}^{n}}dx}=\dfrac{{{x}^{n+1}}}{n+1}$ , where $n\ne -1$. To use this formula for the constant term 3, write it as $3{{x}^{0}}$ 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 $f\left( x \right)=8{{x}^{3}}+5{{x}^{2}}-9x+3$ and we are asked to find its antiderivative, in other words we have to integrate it. Let us assume its integral as I, so we have,
$I=\int{\left( 8{{x}^{3}}+5{{x}^{2}}-9x+3 \right)}dx$
Breaking the integral into four parts, one for each term, we have,
$\Rightarrow I=\int{8{{x}^{3}}dx}+\int{5{{x}^{2}}dx}-\int{9xdx}+\int{3dx}$
Now, we can write the constant term 3 as $3{{x}^{0}}$, so we have,
$\Rightarrow I=\int{8{{x}^{3}}dx}+\int{5{{x}^{2}}dx}-\int{9{{x}^{1}}dx}+\int{3{{x}^{0}}dx}$
Now, applying the basic formula of integral given as: - $\int{{{x}^{n}}dx}=\dfrac{{{x}^{n+1}}}{n+1}$, where $n\ne -1$, we get,

& \Rightarrow I=8\left( \dfrac{{{x}^{3+1}}}{3+1} \right)+5\left( \dfrac{{{x}^{2+1}}}{2+1} \right)-9\left( \dfrac{{{x}^{1+1}}}{1+1} \right)+3\left( \dfrac{{{x}^{0+1}}}{0+1} \right) \\\ & \Rightarrow I=8\left( \dfrac{{{x}^{4}}}{4} \right)+5\left( \dfrac{{{x}^{3}}}{3} \right)-9\left( \dfrac{{{x}^{2}}}{2} \right)+3x \\\ & \Rightarrow I=2{{x}^{4}}+\dfrac{5{{x}^{3}}}{3}-\dfrac{9{{x}^{2}}}{2}+3x \\\ \end{aligned}$$ Taking L.C.M., i.e., 6 and simplifying we get, $$\begin{aligned} & \Rightarrow I=\dfrac{12{{x}^{4}}+10{{x}^{3}}-27{{x}^{2}}+18x}{6} \\\ & \Rightarrow I=\dfrac{1}{6}\left( 12{{x}^{4}}+10{{x}^{3}}-27{{x}^{2}}+18x \right) \\\ \end{aligned}$$ Now, since the given integral was an indefinite integral and therefore we need to add a constant of integration (C) in the expression obtained for I. So, we get, $$\Rightarrow I=\dfrac{1}{6}\left( 12{{x}^{4}}+10{{x}^{3}}-27{{x}^{2}}+18x \right)+C$$ Hence, the above relation is required answer **Note:** One may note that the basic formula that we have used to find the integral I given as: - $$\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..