Question

What is `f(x) = int -9x+5sqrt(x^2+1) dx` if `f(2) = 7`?

Answer 1

`f(x) = -9/2x^2 + 5/2secxtanx +5/2ln|secx + tanx| + 8.06`

Explanation:

Separate the integrals.

`f(x) = int -9xdx + int 5sqrt(x^2 +1)dx`

The first integral is easy, we can do this using `int x^n dx = x^(n + 1)/(n + 1) + C`, where `n != -1`. The second integral, though, will require trig substitution.

Since `int 5sqrt(x^2 + 1)` is of the form `a^2 + x^2`, we use the substitution `x = tantheta`. This means that `dx = sec^2theta d theta`.

`5int sqrt(x^2 + 1)dx = 5int sqrt((tan theta)^2 + 1) * sec^2theta d theta`

`5int sqrt(x^2 + 1)dx = 5int sqrt(sec^2theta) * sec^2theta d theta`

`5int sqrt(x^2 + 1)dx = 5int sec^3theta d theta`

This is a known integral that can be found here.

`5intsqrt(x^2 +1)dx = 5(1/2secxtanx + 1/2ln|secx + tanx|)`

`5intsqrt(x^2 + 1)dx = 5/2secxtanx + 5/2ln|secx + tanx|`

We now put all of this together and add the constant of integration.

`f(x) = -9/2x^2 + 5/2secxtanx +5/2ln|secx + tanx| + C`

We now solve for `C`.

`7 = -9/2(2)^2 + 5/2sec(2)tan(2) + 5/2ln|sec2 + tan2| + C`

Using a calculator, we get an approximation of `8.0648~~ 8.06`.

Hopefully this helps!