Question

If f'(x)=x²+5, then ∫ f(x) dx=.........(c and k are arbitary constants),Select Proper option from the given options.
(a) x⁴/12 + 5x²/8 +cx+k
(b) - x⁴/12 - 5x²/2 -cx+k
(c) x⁴/12 - 5x²/12 +cx+k
(d) x⁴/12 + 5x²/2 +cx+k

Answer 1

Answer:

∫ f(x) dx = x^4 / 12 + 5x²/2 + cx + k

Thus option D is correct

Step-by-step explanation:

f'(x) = x² + 5

Then integrating on both side we get

f(x) = ∫( x² + 5 )dx

     = ∫x² dx + ∫5 dx

     = x³/3 + 5x + c            where c is the constant of integration

Now

f(x) = x³/3 + 5x + c  

So

Integrating on both sides again we get

∫ f(x) dx = ∫(x³/3 + 5x + c) dx

             = ∫(x³/3) dx + ∫5x dx + ∫c dx

             = x^4 / 12 + 5x²/2 + cx + k       where k is constant of integration

Hence

∫ f(x) dx = x^4 / 12 + 5x²/2 + cx + k

Thus option D is correct