Chapter 4 - Integration - 4.9 Evaluating Definite Integrals By Substitution - Exercises Set 4.9 - Page 342: 50

See explanation.

Left side equation \[ \int_{0}^{t} (-x+t)f g(x) d x \] \[ -\int_{t}^{0} f(u) (-u+t)g d u \] Substitution $-x+t=u$ Replace $(d u)\ with\ (-d x)$ \[ \int_{0}^{t} f(u) g(t-u) d u \] Rewrite integral \[ \int_{0}^{t} f(x) (-x+t) g d u \]