What is f(x) = int x-xsecx dx if f((7pi)/8) = 0 ?

1 Answer
Mar 9, 2017

f(x)=1/2x^2-xln|secx+tanx|-secx-3.7522

Explanation:

f(x)=int(x-xsecx)dx

f(x)=int(x)dx-color(blue)(int(xsecx)dx)

Use integration by parts to simplify: color(blue)(int(xsecx)dx)
color(blue)(u=x " " v = ln|secx+tanx|)
color(blue)(du=dx" "dv=secx)
color(blue)(int(xsecx)dx=xln|secx+tanx|-int(ln|secx+tanx|)dx)
color(blue)(=xln|secx+tanx|-secx+C_0)

f(x)=1/2x^2-xln|secx+tanx|-secx+C

Given: f((7pi)/8)=0
0=1/2((7pi)/8)^2-((7pi)/8)ln|sec((7pi)/8)+tan((7pi)/8)|-sec((7pi)/8)+C

Since this is difficult to simplify, the rest of this response consists of approximations of C (using calculator).
Capprox-3.752247019

f(x)=1/2x^2-xln|secx+tanx|-secx-3.7522