几道积分题,∫x2lnxdx∫xcosx/2dx∫arccosxdx∫xe-xdx∫lnx/√2dx那个－x是在e的上面的∫x2lnxdx中的2是平方的意思

Question

几道积分题,
∫x2lnxdx
∫xcosx/2dx
∫arccosxdx
∫xe-xdx
∫lnx/√2dx
那个－x是在e的上面的
∫x2lnxdx
中的2是平方的意思

Answer 1

∫x^2*lnxdx
=∫lnxd1/3(x^3)
=1/3*x^3*lnx-∫x^3/3dlnx
=1/3*x^3*lnx-∫x^3/3*(1/x)dx
=1/3*x^3*lnx-∫x^2/3dx
=1/3*x^3*lnx-x^3/9+c
∫xcos(x/2)dx
=∫xd(2sinx/2)
=2x*sin(x/2)-∫cos(x/2)dx
=2x*sin(x/2)-2sin(x/2)+c
∫arccosxdx
=x*arccosx-∫xdarccosx
=x*arccosx+∫x*1/(√(1+x^2))dx
=x*arccosx+∫1/(√(1+x^2))d(x^2)/2
=x*arccosx+1/√(1+x^2)*x^2/2-∫x/√(1+x^2)dx
=x*arccosx+x^2/(2*√(1+x^2))-(1/2)*√(1+x^2)+c
∫x^e-xdx
=∫x^edx-∫xdx
=*x^(e+1)/(e+1)-(x^2/2)+c
∫lnx/√2dx
=(√2/2)*∫lnxdx
=(√2/2)*(xlnx-∫xdlnx)
=(√2/2)*(xlnx-∫1dx)
=(√2/2)*(xlnx-x)+c