x加根号下1加x的平方的原函数是什么意思

Answer 1

x+√(1+x²)的原函数,即对x+√(1+x²)进行不定积分
∫[x+√(1+x²)]dx
=½x²+∫√(1+x²)dx
∫√(1+x²)dx
=x√(1+x²)-∫xd[√(1+x²)]
=x√(1+x²)-∫[x²/√(1+x²)]dx
=x√(1+x²)-∫[(x²+1-1)/√(1+x²)]dx
=x√(1+x²)-∫√(1+x²)dx+∫dx/√(1+x²)
=x√(1+x²)-∫√(1+x²)dx+ln|x+√(1+x²)|
∴∫√(1+x²)dx=½[x√(1+x²)+ln|x+√(1+x²)|]
∴x+√(1+x²)的原函数=½[x²+x√(1+x²)+ln|x+√(1+x²)|]+C

Answer 2

69、浪淘沙 刘禹锡