Using a Given Substitution

A supplied substitution can simplify an integral that is not immediately in reverse-chain-rule form. Rewrite the integrand, differential and bounds in the new variable before integrating. Use the common “Reverse Chain Rule and Substitution” card for direct reverse-chain-rule patterns; here the substitution is supplied and must be applied to the whole integral. Example Use x=2sintheta, with 0 <= theta <= 2, to evaluate _01sqrt(4-x2),dx. dx=2costheta,dtheta. The bounds x=0 and x=1 give theta=0 and theta= 6, respectively. On this interval costheta 0, so sqrt(4-x2)=sqrt(4cos2theta)=2costheta. _01sqrt(4-x2),dx= _0(pi)/(6)4cos2theta,dtheta = _0(pi)/(6)2(1+cos2theta),dtheta =[2theta+sin2theta]_0(pi)/(6) = 3+ 32. For a definite integral, either use transformed bounds throughout or return to the original variable before applying the original bounds. For an indefinite integral, substitute back and include an arbitrary constant. Choose an interval on which the substitution is one-to-one and matches the original domain. In general, sqrt(a2cos2theta)=a|costheta| for a>0; remove the modulus only after checking the sign. A decreasing substitution reverses the order of the transformed bounds. Keep that order: the differential accounts for the direction change.