3 Calculate for Hint: Try substitution. Do not use technology! Note: Substitution is the inverse operation of the chain rule. We know that f(g(x)) = f'(g(x)) g' (r). The substitution rule derives from applying the Fundamental Theorem of Calculus to this formula. Indeed, the FTC will tell us that f(g(x)) + C = f f' (g(x)) g' (x) dr. The trick consists in writing the integrand in our problem as the right hand side in this equal- ity. Precisely, given f h(r)dx, we try to re-write h as h(x) = k(u(x))u'(x). If K' = k, the formula tells us that fh(r)dx = fK'(u(x))u'(x)dr = K (u(x)) + C. When applied to definite integrals, it is common to short- cut the answer by avoiding to go back to functions of x, by adjusting the integration limits in, say, h(r)da, to the limits in the substitution func- tion u, that is, writing the integral as fu) K'(u)du. The solutions below adopt this shortcut (it wasn't popular in your answers, but it is worth remembering as a simplifying option of course, it's not mandatory). mit, of course).

Fig: 1