Question

高数-求不定积分

怎么求？

Answer 1

∵反双曲正弦函数(arcsinhx)'=1/√（1+x^2),
∴∫dx/√（1+x^2)=arcsinhx+C=ln[x+√（1+x^2)]+C,
若不用反双曲函数，则用第二类换元法
x=tant
t=arctanx
sect=√(x²+1)
1+x²=(sect)^2,
dx=(secx)^2,
原式=∫(sect)^2dt/sect
=∫sectdt
=∫(1/cost)dt
=∫[cost/(cost)^2]dt
=∫d(sint)/[1-(sint)^2
=(1/2)∫[1/(1-sint)+1/(1+sint)]d(sint)
=(1/2)[ln|1+sint|-ln|1-sint|]+C
=(1/2)ln|(1+sint)/(1-sint)|+C
=(1/2)ln(sect+tant)/(sect-tant)+C
=ln|sect+tant|+C
=ln(x+√(x²+1))+C ,
倒数第三步，分子分母同乘sect+tant，分母=(sect)^2-(tant)^2=1,

Answer 2

令x=tana
a=arctanx
seca=√(x²+1)
1+x²=sec²a
dx=sec²ada
原式=∫sec²ada/seca
=∫secada
=∫(1/cosa)da
=∫[cosa/cos²a]da
=∫d(sina)/(1-sin²a)
=(1/2)∫[1/(1-sina)+1/(1+sina)]d(sina)
=(1/2)[-ln|1-sina|+ln|1+sina|]+C
=(1/2)ln|(1+sina)/(1-sina)|+C
=ln|seca-tana|+C
=ln|√(x²+1)-x|+C