Question

两题不定积分 ∫x(secx^2)tanx dx ∫x^(1/2)lnx dx

Answer 1

∫x(secx^2)tanx dx
=∫xtanxdtanx
=x(tanx)^2-∫tanxdxtanx
=x(tanx)^2-∫[(tanx)^2+xtanx(secx)^2]dx
=x(tanx)^2-∫[(secx)^2-1]dx-∫x(secx)^2tanxdx
=x(tanx)^2-tanx+x-∫x(secx)^2tanxdx
所以∫x(secx^2)tanx dx =(1/2)*{x(tanx)^2-tanx+x}+c
∫x^(1/2)lnx dx
=(2/3)*∫lnxdx^(3/2)
=(2/3)*[x^(3/2)*lnx-∫x^(3/2)*1/xdx]
=(2/3)*[x^(3/2)*lnx-∫x^(1/2)dx]
=(2/3)*[x^(3/2)*lnx-(2/3)*x^(3/2)]+c
=(2/3)*[x^(3/2)*lnx]-(4/9)*x^(3/2)+c