微积分 求步骤

Question

如题所述

Answer 1

1、
原式=∫(π/3,π)sin(x+π/3)d(x+π/3)
=-cos(x+π/3)|(π/3,π)
=cos(π/3+π/3)-cosπ
=-1/2-(-1)
=1/2
2、
原式=1/2∫(0,√ln2)x^2e^(x^2)d(x^2)
=1/2∫(0,√ln2)x^2de^(x^2)
=1/2x^2e^(x^2)|(0,√ln2)-1/2∫(0,√ln2)2xe^(x^2)dx
=1/2[(√ln2)^2e^(√ln2)^2-0]-1/2∫(0,√ln2)e^(x^2)d(x^2)
=(1/2)ln2e^(ln2)-1/2e^(x^2)|(0,√ln2)
=ln2-1/2[e^(√ln2)^2-e^(0^2)]
=ln2-1/2[2-1]
=ln2-1/2
3、
易知，被积函数为奇函数，而奇函数在对称区间的积分为0。
因此，原式=0