Question

Find f(x) if f'(x)= `(e^(2x) +4e^-x)/e^x` and f(ln2)=2. Do u just find the anti-derivative for numerator and denominator separately?

Answer 1

No you can't do it that way.

Explanation:

The derivative of a quotient is not just the quotient of the derivatives, so, likewise, the anti derivative of a quotient is not just the quotient of antiderivatives..

In general, finding antiderivatives is much harder than finding derivatives. (In fact some antiderivatives cannot be nicely expressed.)

Noe way to find the antiderivative of

f'(x)= `(e^(2x) +4e^-x)/e^x`

is to rewrite it.

f'(x)= `(e^(2x) +4e^-x)/e^x = e^(2x)/e^x + (4e^(-x))/e^x`

`= e^x + 4e^(-2x)`

Now, the antiderivative of a sum is just the sum of the antiderivative (just as the derivative of a sum is he sum of the derivatives).

So,

`f(x) = e^x -2e^(-2x) +C`

(Check the answer by differentiating.)

Now use the given point to find `C`.