不定积分计算题
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(1) ∫x^2/(x^2+1)^2 dx =∫ dx/(x^2+1) - ∫ dx/(x^2+1)^2 =arctanx - ∫dx/(x^2+1)^2 let x= tany dx= (secy)^2 dy ∫dx/(x^2+1)^2 =∫dy/(secy)^2 =∫(cosy)^2 dy =(1/2)∫(1+cos2y) dy =(1/2)[y + (1/2)sin2y] + C' =(1/2)[arctanx + x/(1+x^2)] + C' ∫x^2/(x^2+1)^2 dx =arctanx - ∫dx/(x^2+1)^2 =arctanx - (1/2)[arctanx + x/(1+x^2)] + C =(1/2)[arctanx - x/(1+x^2)] + C (2) t^5 +1 = t^4(t+1) - t^4 +1 =t^4(t+1) - t^3(t+1) + t^3 +1 =t^4(t+1) - t^3(t+1) + t^2(t+1) -t^2 +1 =t^4(t+1) - t^3(t+1) + t^2(t+1) -t(t+1)+ t+1 =t^4(t+1) - t^3(t+1) + t^2(t+1) -t(t+1)+ (t+1) ie t^5 +1 = (t+1)(t^4-t^3+t^2-t+1) (t^5+1)/(t+1) =t^4-t^3+t^2-t+1 ∫ (t^5+1)/(t+1) dt =∫( t^4-t^3+t^2-t+1) dt =(1/5)t^5-(1/4)t^4+(1/3)t^3-(1/2)t^2 +t + C
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