Question

大神知道这个积分题怎么做吗？

Answer 1

令x=sht(双曲正弦)
原式= ∫ch²tdt =t/2+1/4∫ch2td(2t)
=t/2+sh2t/4+C =ln(x+√(1+x²))/2+x√(1+x²)/2+C
令x=tant，-π/2<t<π/2
原式= ∫sec^3tdt =∫sectdtant =secttant-∫tan^2tsectdt
=secttant-∫(sec^2t-1)sectdt =secttant-∫sec^3tdt+∫sectdt
把-∫sec^3tdt移项到等式左边 2∫sec^3tdt=secttant+∫sectdt =secttant+∫sect(sect+tant)/(sect+tant)dt
=secttant+∫d(sect+tant)/(sect+tant) =secttant+ln(sect+tant)+C' =x√(1+x^2)+ln(x+√(1+x^2))+C'

左右除以2，即可得