Question

5.
Find the unique antiderivative F(x) of f(x)=2x2+1, whese F(0) = -2.


find this​

Answer 1

Correct Question:-

Find the unique anti - derivative $F(x)$ of $f(x)=2x^2+1$ where $F(0)=-2.$

Solution:-

Anti - derivative of a function in $x$ is nothing but it's integral with respect to $x.$

Given that $F(x)$ is the anti - derivative of $f(x).$ So,

$\displaystyle\longrightarrow F(x)=\int f(x)\ dx$

But here, $f(x)=2x^2+1.$ Then,

$\displaystyle\longrightarrow F(x)=\int\left(2x^2+1\right)\ dx$

Providing integral to each term,

$\displaystyle\longrightarrow F(x)=\int 2x^2\ dx+\int 1\ dx$

The constants can be taken out of the integral.

$\displaystyle\longrightarrow F(x)=2\int x^2\ dx+1\int dx$

$\displaystyle\longrightarrow F(x)=2\int x^2\ dx+\int x^0\ dx\quad\quad\dots(1)$

We know that,

• $\displaystyle \int x^n\ dx=\dfrac{x^{n+1}}{n+1},\quad n\neq-1$

Hence,

$\displaystyle\longrightarrow F(x)=2\cdot \dfrac{x^{2+1}}{2+1}+\dfrac{x^{0+1}}{0+1}+c$

$\displaystyle\longrightarrow F(x)=2\cdot \dfrac{x^{3}}{3}+\dfrac{x^{1}}{1}+c$

$\displaystyle\longrightarrow F(x)=\dfrac{2}{3}\,x^3+x+c\quad\quad\dots(2)$

But given that,

$\displaystyle\longrightarrow F(0)=-2$

$\displaystyle\longrightarrow \dfrac{2}{3}\,(0)^3+0+c=-2$

$\displaystyle\longrightarrow c=-2$

Hence (2) becomes,

$\displaystyle\longrightarrow\underline{\underline{F(x)=\dfrac{2}{3}x^3+x-2}}$