Question

If f'(x)=8x³-2x,f(2)=8, then find f(x).Integrate the given function w.r.t. x considering them well defined and integrable over proper domain.

Answer 1

In the attachment I have answered this problem.           See the attachment for detailed solution.

Answer 2

Dear Student,

Solution:

As f'(x) is given and we have to calculate f(x)

Since we know that integration is the reverse process of differentiation.

So, if we integrate f'(x) ,we will get f(x)

Now integration of
${x}^{n} = \frac{ {x}^{n + 1} }{n + 1}$
provided that n ≠ -1

So, to convert f'(x)=8x³-2x into f(x), integrate it

$f(x) = 8 \times \frac{ {x}^{4} }{4} - 2 \times \frac{ {x}^{2} }{2} + c \\ \\ f(x) = 2 {x}^{4} - {x}^{2} + c$
since given that
$f(2) = 8$
put this value in last equation,to find the value of c

$8 = 2( {2)}^{4} - ( {2})^{2} + c \\ 8 = 32 - 4 + c \\ c = 8 + 4 - 32 \\ c = - 20$
So,
$f(x) = 2 {x}^{4} - {x}^{2} - 20$
is the value of f(x).

Hope it helps you...