Question

What is `f(x) = int 5e^(2x)-2e^x+3x dx` if `f(4 ) = 2`?

Answer 1

This can be broken down into separate integrals.

`f(x) = int(5e^(2x) - 2e^x + 3x)dx`

`f(x) = int(5e^(2x))dx + int(-2e^x)dx + int(3x)dx`

We know that `int(ae^(nx))dx = a/n e^(nx) + C` and `int(x^n)dx = (x^(n + 1))/(n + 1) + C`. Therefore:

`f(x) = 5/2e^(2x) - 2e^x + 3/2x^2 + C`

We can now solve for `C`.

`2 = 5/2e^(8) - 2e^(4) + 3/2 4^2 + C`

`2 = 5/2e^8 - 2e^4 + 24 + C`

`2 - 5/2e^8 - 2e^4 - 24 = C`

`C = -22 - 5/2e^8 - 2e^4`

This is as simplified a form as we can get without using a calculator to approximate.

Use any calculator to get `C ~= -131.197`.

Hence, `f(x) = 5/2e^(2x) - 2e^x + 3/2x^2 - 131.197`.

Hopefully this helps!