Finding the antiderivative of a function

Calculus Level 4

Let f ( x ) = x 2 + 4 x 3 f(x) = \sqrt[3]{x^2+4x} , and let g ( x ) g(x) be an antidervative of f ( x ) f(x) . If g ( 5 ) = 7 g(5) = 7 , find the value of g ( 1 ) g(1) .

Give your answer to 3 decimal places.


The answer is -3.88222.

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Hi Hi by Hi Hi , 18, USA

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1 solution

Hi Hi
Jul 2, 2016

Since g(x) is an antiderivative of f(x), we have 

    g '(x)  =  f(x)

or

None of the regular techniques of integration will work on this integral. Even the computer cannot solve this explicitly. Instead of integrating, we let 

    h(x)  =  g(x) - 7

Then h(x) is also an antiderivative of f(x) and 

    h(5)  =  0

We can write

Notice that when we plug in 5 for x, we get 0 as required, since the upper and lower limits are equal. Now use a calculator to easily find

finally since 

    h(x)  =  g(x) - 7

it follows that 

    g(x)  =  h(x) + 7

and that 

    g(1)  =  h(1) + 7  =  -10.88222 + 7  =  -3.88222
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Why calculator ???

Give a mathematical way out

Kushal Bose - 4 years, 11 months ago

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Calculators are for the lazy... but I was too lazy ;) There definitely is a mathematical way out, but I sorta didn't have time to put it in, when I have time I'll do it :P

Hi Hi - 4 years, 11 months ago

@Kushal Bose This is the reason I used a calculator... even though with 10 minutes I could calculate it, I'd rather just use a calculator ;)

Hi Hi - 4 years, 11 months ago

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