Calculus: Early Transcendentals, Chapter 7, 7.1, Section 7.1, Problem 6

You need to solve the integral int (x-1) sin (pi*x) dx = int x*sin (pi*x) dx - int sin (pi*x)dx
You need to use substitution pi*x = t => pi*dx = dt => dx = (dt)/(pi)
int x*sin (pi*x) dx = 1/(pi^2) int t*sin t
You need to use the integration by parts for int t*sin t such that:
int udv = uv - int vdu
u = t => du = dt
dv = sin t=> v = -cos t

int t*sin t = -t*cos t + int cos t dt
1/(pi^2) int t*sin t = 1/(pi^2)(-t*cos t + sin t) + c
Replacing back the variable yields:
int x*sin (pi*x) dx = 1/(pi^2)(-pi*x*cos(pi*x) + sin (pi*x)) + c
int (x-1) sin (pi*x) dx = 1/(pi^2)(-pi*x*cos(pi*x) + sin (pi*x))+ (cos (pi*x))/(pi) + c
Hence, evaluating the integral, using integration by parts, yields int (x-1) sin (pi*x) dx = 1/(pi^2)(-pi*x*cos(pi*x) + sin (pi*x))+ (cos (pi*x))/(pi) + c.