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Find a function fthat has the given derivative f'and value f(c).

f'x=x1+4x2,f0=1

Short Answer

Expert verified

The anti derivative is fx=18ln1+4x2+1

Step by step solution

01

Step 1:Given information

The given expression isf'x=x1+4x2

02

Step 2:Simplification

Given

f'x=x1+4x2

We will use an intelligent guess-and-check method to find f

Since differentiating a power function one and derivative of lnxis1x

One can start with fx=ln1+4x2

Now,

f'x=ddxln1+4x2

=11+4x2ddx1+4x2

=8x1+4x2

Hence, we can guess the function as

fx=18ln1+4x2+c

Now ,

f0=11=ln(1+0)+c

c=1

Therefore,

fx=18ln1+4x2+1

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