Question

定积分问题，两道题

(1) ∫(上限e，下限1/2)绝对值lnxdx
(2) ∫(上限π，下限0)x平方sinxdx
要详细过程，谢谢啦

Answer 1

Answer 2

(1)
∫(1/2->e) |lnx| dx
= ∫(1/2->1) -lnx dx + ∫(1->e) lnx dx
= [-xlnx + x](1/2->1) + [xlnx-x](1->e)
= 1+(1/2)ln(1/2) -1/2 + e - e +1
= 3/2 -(1/2)ln2
(2)
∫(0->π)x^2 sinx dx
=-∫(0->π)x^2 dcosx
= - [x^2cosx](0->π) +2∫(0->π)xcosxdx
= π^2+2∫(0->π)xdsinx
= π^2+ 2[xsinx](0->π) - 2∫(0->π)sinx dx
=π^2+ 2[cosx](0->π)
=π^2 -4