灵鹊做题网求ln(1+x)/√xdx的不定积分 还有一题xarctanxdx的不定积分
来源：学生作业帮 编辑：灵鹊做题网作业帮 分类：数学作业 时间：2024/04/25 07:57:33
求ln(1+x)/√xdx的不定积分 还有一题xarctanxdx的不定积分
∫ ln(1+x)/√x dx
= 2∫ ln(1+x)/(2√x) dx
= 2∫ ln(1+x) d√x
= 2ln(1+x) * √x - 2∫ √x dln(1+x),integration by part
= 2(√x)ln(1+x) - 2∫ √x/(1+x) dx
= 2(√x)ln(1+x) - 2∫ (2√x*√x)/[2√x*(1+x)] dx
= 2(√x)ln(1+x) - 4∫ [1+(√x)²-1]/[1+(√x)²] d√x
= 2(√x)ln(1+x) - 4∫ d√x + 4∫ d√x/[1+(√x)²]
= 2(√x)ln(1+x) - 4√x + 4arctan(√x) + C
∫ xarctanx dx
= ∫ arctanx d(x²/2)
= (1/2)x²arctanx - (1/2)∫ x² d(arctanx),integration by part
= (1/2)x²arctanx - (1/2)∫ x²/(1+x²) dx
= (1/2)x²arctanx - (1/2)∫ (1+x²-1)/(1+x²) dx
= (1/2)x²arctanx - (1/2)∫ dx + (1/2)∫ dx/(1+x²)
= (1/2)x²arctanx - x/2 + (1/2)arctanx + C
= 2∫ ln(1+x)/(2√x) dx
= 2∫ ln(1+x) d√x
= 2ln(1+x) * √x - 2∫ √x dln(1+x),integration by part
= 2(√x)ln(1+x) - 2∫ √x/(1+x) dx
= 2(√x)ln(1+x) - 2∫ (2√x*√x)/[2√x*(1+x)] dx
= 2(√x)ln(1+x) - 4∫ [1+(√x)²-1]/[1+(√x)²] d√x
= 2(√x)ln(1+x) - 4∫ d√x + 4∫ d√x/[1+(√x)²]
= 2(√x)ln(1+x) - 4√x + 4arctan(√x) + C
∫ xarctanx dx
= ∫ arctanx d(x²/2)
= (1/2)x²arctanx - (1/2)∫ x² d(arctanx),integration by part
= (1/2)x²arctanx - (1/2)∫ x²/(1+x²) dx
= (1/2)x²arctanx - (1/2)∫ (1+x²-1)/(1+x²) dx
= (1/2)x²arctanx - (1/2)∫ dx + (1/2)∫ dx/(1+x²)
= (1/2)x²arctanx - x/2 + (1/2)arctanx + C
求ln(1+x)/√xdx的不定积分 还有一题xarctanxdx的不定积分
求∫√1-x² 分之xdx的不定积分
求不定积分∫x^2 ln xdx
求(ln√x)^2的不定积分
ln(x+1)的不定积分?
求不定积分 arcsinx的不定积分 e^√x+1的不定积分 (x-1)lnx的不定积分
求不定积分∫xdx/√3x^2-1,
求不定积分∫dx/x√1-ln^2 x 是ln平方的x
求[x+ln(1-x)]/x^2的不定积分~
求：∫lnx/根号xdx的不定积分,答案说等于：4根号x((ln根号x)-1)+c,是怎么得的.
(ln x)/x+1的不定积分怎么求?
ln√x/x的不定积分是多少