What is f(x) = int -9x+5sqrt(x^2+1) dx if f(2) = 7 ?
1 Answer
Explanation:
Separate the integrals.
f(x) = int -9xdx + int 5sqrt(x^2 +1)dx
The first integral is easy, we can do this using
Since
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
7 = -9/2(2)^2 + 5/2sec(2)tan(2) + 5/2ln|sec2 + tan2| + C
Using a calculator, we get an approximation of
Hopefully this helps!