求下列不定积分2“——”前为题目,后为该题答案,我没有凑出来答案数.1.∫xe^(3x^2)dx——1/6(e^3x)^2+C2.∫xcos[(x^2)+1]dx——1/2sin(x^2+1)+C3.∫x^2/(1+x^3)dx——1/3ln|1+x^3|+C4.∫(x+1+arctanx)/(1+x^2)dx—

Question

求下列不定积分2
“——”前为题目,后为该题答案,我没有凑出来答案数.
1.∫xe^(3x^2) dx ——1/6(e^3x)^2+C
2.∫xcos[(x^2)+1] dx ——1/2sin(x^2 +1)+C
3.∫x^2/(1+x^3) dx ——1/3ln|1+x^3|+C
4.∫(x+1+arctanx)/(1+x^2) dx ——1/2ln(1+x^2)+arctanx+1/2(arctanx)^2+C
5.∫1/[x(4+lnx)] dx ——ln|4+lnx|+C
6.∫1/(e^x)+(e^-x) dx ——arctane^x+C

Answer 1

1.∫xe^(3x²) dx
= ½∫e^(3x²) dx²
= (1/6)∫e^(3x²) d(3x²)
= (1/6)∫de^(3x²)
= 1/6(e^3x)²+C
2.∫xcos(x²+1)dx
=½∫cos(x²+1)dx²
=½∫cos(x²+1)d(x²+1)
=½ sin(x² +1)+C
3.∫x²/(1+x³)dx
= ⅓∫1/(1+x³)dx³
= ⅓∫1/(1+x³)d(x³+1)
= ⅓ ln|1+x³|+C
4.∫(x+1+arctanx)/(1+x²)dx
=∫x/(1+x²)dx+∫1/(1+x²)dx+∫arctanx)/(1+x²)dx
=½∫1/(1+x²)dx²+∫1/(1+x²)dx+∫arctanx)d(arctanx)
=½∫1/(1+x²)d(x²+1)+∫1/(1+x²)dx+∫arctanx)d(arctanx)
=½ln|x²+1|+arctanx+½(arctanx)²+C
5.∫1/[x(4+lnx)]dx
= ∫1/(4+lnx)d(lnx)
= ∫1/(4+lnx)d(4+ln|x|)
= ln|4+ln|x||+C
6.∫1/[(e^x)+(e^-x)]dx
= ∫e^x/{e^x[(e^x)+(e^-x)]}dx
= ∫e^x/[(e^2x)+1]dx
= ∫1/[(e^2x)+1]d(e^x)
= arctane^x+C