Question

求下列不定积分：

求下列不定积分：
1)In(x平方+ 1)dx=
2)x的n次方Inxdx(n不等于1)
3)X三次方（Inx）的平方dx
要详细步骤！！谢谢各位了，拜托了

Answer 1

∫ln(x²+1)dx
=xln(x²+1)-∫xdln(x²+1)
=xln(x²+1)-2∫x²/(x²+1)dx
=xln(x²+1)-2∫[(x²+1)-1]/(x²+1)dx
=xln(x²+1)-2∫dx+2∫1/(x²+1)dx
=xln(x²+1)-2x+2arctanx+C

令lnx=t，x=e^t，dx=e^tdt

∫(x^n)lnxdx
=∫t(e^t)[(e^t)^n]dt
=∫te^[(n+1)t]dt
={∫(n+1)te^[(n+1)t]d(n+1)t}/(n+1)²
={∫(n+1)tde^[(n+1)t]}/(n+1)²
=(n+1)te^[(n+1)t]/(n+1)²-{∫e^[(n+1)t]d(n+1)t}/(n+1)²
=te^[(n+1)t]/(n+1)-e^[(n+1)t]/(n+1)²+C
=[x^(n+1)]lnx/(n+1)-[x^(n+1)]/(n+1)²+C

令lnx=t，x=e^t，dx=e^tdt

∫x³ln²xdx
=∫t²e^(3t)e^tdt
=∫t²e^(4t)dt
=[∫(4t)²e^(4t)d4t]/64
=[∫(4t)²de^(4t)]/64
=(4t)²e^(4t)/64-[∫e^(4t)d(4t)²]/64
=(4t²)e^(4t)/64-[∫(4t)e^(4t)d4t]/32
=(4t)²e^(4t)/64-[∫(4t)de^(4t)]/32
=(4t)²e^(4t)/64-4te^(4t)/32+[∫e^(4t)d4t]/32
=(4t)²e^(4t)/64-te^(4t)/8+e^(4t)/32+C
=(x^4)(4lnx)²/64-(x^4)lnx/8+(x^4)/32+C
=(x^4)ln²x/4-(x^4)lnx/8+(x^4)/32+C

Answer 2

1)ln(x平方+1)dx=(2x)/(x平方+1)