Question

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

Answer 1

`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`