Question

高等数学 大学数学 微积分 求解！！！

Answer 1

I = ∫<-1, 1>xdx/(1+x^2) + ∫<-1, 1>|x|dx/(1+x^2)
= 0 + 2∫<0, 1>xdx/(1+x^2) = ∫<0, 1>d(1+x^2)/(1+x^2)
= [ln(1+x^2)]<0, 1> = ln2
2. I = ∫<1, 2>e^(1/x)dx/x^2 = -∫<1, 2>e^(1/x)d(1/x)
= -[e^(1/x)]<1, 2> = e - √e
3. I = ∫<0, 1>xe^(2x)dx = (1/2)∫<0, 1>xde^(2x)
= (1/2)[xe^(2x)]<0, 1> - (1/2)∫<0, 1>e^(2x)dx
= e^2/2 - (1/4)[e^(2x)]<0, 1> = (1/4)(1+e^2)
4. 联立解 xy = 2, y= 1+x^2 得交点 (1, 2). 则
I = ∫<1, 2>x^2dx ∫<2/x, 1+x^2>dy/y^2
= ∫<1, 2>x^2dx[-1/y]<2/x, 1+x^2>
= ∫<1, 2>x^2[x/2-1/(1+x^2)]dx
= ∫<1, 2>[x^3/2-x^2/(1+x^2)]dx
= [x^4/8 - x + arctanx]<1, 2>
= 7/8 - π/4 +arctan2