What is f(x) = int xsinx- sec2x dx if f(pi/12)=-2 ?

1 Answer

color(blue)(f(x)=-x*cos x+sin x-1/2 ln (sec 2x+tan 2x) + 1/2 ln sqrt3+pi/24*sqrt(2+sqrt3)-1/2*sqrt(2-sqrt3)-2)

also color (blue)(f(x)=-x*cos x+sin x-1/2 ln (sec 2x+tan 2x) -1.73129)

Explanation:

Start by performing color(blue)(" Integration by parts") using the formula

color(blue)(int u dv=uv-int v du )

the solution is quite long because it consists of two terms.

from the given:

int (x sin x - sec 2x) dx

color (red)("First part:")

int (x sinx) dx

Let u=x, and dv=sin x dx
and v=-cos x and du=dx

so that int (x sinx) dx=-x*cos x-int -cos x dx

then color (blue)(int (x sinx) dx=-x*cos x+sin x)

color (red)(" second part:")

Use the formula: int (sec u) du=ln (sec u+tan u)+C

so that

color(blue)(int sec 2x dx=1/2*ln(sec 2x+tan 2x))

Now , combine the first and second parts

color (magenta)(f(x)=int (x sin x - sec 2x) dx=
color(magenta)(-x*cos x+sin x-1/2*ln(sec 2x+tan 2x)+C)

Solve for C: using f(pi/12)=-2 and then

-2=-(pi/12)*cos (pi/12)+sin (pi/12)-1/2*lnabs(sec 2*(pi/12)+tan 2*(pi/12))+C

-2=-(pi/12)*1/2*sqrt(2+sqrt3)+1/2*sqrt(2-sqrt3)-1/2*lnabs(2/sqrt3+1/sqrt3)+C

C=1/2 ln sqrt3+pi/24*sqrt(2+sqrt3)-1/2*sqrt(2-sqrt3)-2

The final answer is

color (magenta)(f(x)=-x*cos x+sin x-1/2 ln (sec 2x+tan 2x) + 1/2 ln sqrt3+pi/24*sqrt(2+sqrt3)-1/2*sqrt(2-sqrt3)-2)

simplifying the square roots we have the following

color (magenta)(f(x)=-x*cos x+sin x-1/2 ln (sec 2x+tan 2x) -1.73129)