Calculus: Early Transcendentals, Chapter 7, 7.1, Section 7.1, Problem 8

You need to use the substitution beta*t = u , such that:
beta*t = u => beta dt = du
Replacing the variable, yields:
int t^2*sin(beta*t) dt = 1/(beta^3) int u^2*sin u du
You need to use the integration by parts such that:
intfdg =fg - int gdf
f =u^2 => df = 2udu
dg =sin u=>g = -cos u
1/(beta^3)int u^2*sin u du = 1/(beta^3)(-u^2*cos u + 2int u*cos u du)
You need to use the integration by parts again, such that:
2int u*cos u du = 2u*sin u - 2int sin u du
f =u => df = du
dg =cos u=>g = sin u
2int u*cos u du = 2u*sin u + 2cos u + c
1/(beta^3)int u^2*sin u du = 1/(beta^3)(-u^2*cos u + 2u*sin u + 2cos u) + c
Replacing back the variable, yields:
int t^2*sin(beta*t) dt = 1/(beta^3)(-(beta*t)^2*cos(beta*t) + 2(beta*t)*sin(beta*t) + 2cos (beta*t)) + c
Hence, evaluating the integral, using substitution, then integration by parts, yields int t^2*sin(beta*t) dt = 1/(beta^3)(-(beta*t)^2*cos(beta*t) + 2(beta*t)*sin(beta*t) + 2cos (beta*t)) + c