Question

几道微积分题目 （1）∫X^2/(根号1-X^2)dx (2)∫ln(1+X)dx (3)∫x*cos平方Xdx

Answer 1

（1）令x=sint,因x属于(-1,2),故t在(-pi/2,pi/2)内,且dx=costdt ∫x^2/根号(1-x^2)dx =∫(sint)^2/cost×costdt =∫(sint)^2 dt =∫(1-cos2t)/2 dt =t/2-sin2t/4+C =arcsinx/2-x×根号(1-x^2)/2+C (2)∫ln(1+x)dx =∫ln(x+1)d(x+1) =(x+1)ln(x+1)-∫(x+1)dln(x+1) =(x+1)ln(x+1)-∫1 dx =(x+1)ln(x+1)-x+C (3)∫x*cos平方xdx =∫x(1+cos2x)/2 dx =∫x/2 dx+ ∫xcos2x/2 dx =x^2/4+x sin2x/4-∫sin2x/4 dx =x^2/4+x sin2x/4+cos2x/8 + C