我再问大学里3个数学题：1，∫(arccosx)2 dx= 2，∫xarccosx/(1-x2 )2 dx= 3，∫xarctanx/(1+x2 )2/5 dx= 求他们的不

我再问大学里3个数学题：1，∫(arccosx)2 dx= 2，∫xarccosx/(1-x2 )2 dx= 3，

∫xarctanx/(1+x² )^2/5 dx= 它们的

不定积分是什么呀，急急

t=arccosx

1. ∫(arccosx)^2 dx=∫t^2 d cost=t^2 cost-∫cost d t^2=t^2 cost-2∫tcost dt=t^2 cost-2∫t d sint

=t^2 cost-2(tsint-∫sint dt)=t^2 cost-2tsint-2cost-C

2. ∫xarccosx/(1-x2 )2 dx=∫tcost/(sin^4 t)d cost=-∫tcost/(sin^3 t) dt=-∫t/(sin^3 t) d sint=(1/2)∫t d(csc^2 t)

=(1/2)(tcsc^2 t-∫csc^2 t dt)=(1/2)(tcsc^2 t+cot t)-C

3. t=arctanx

∫xarctanx/(1+x² )^2/5 dx=∫t tant/(sec^3 t)dt=-∫t(cos^2 t)dcost=(-1/3)∫td(cos^3 t)=(1/3)(∫cos^3 t dt-tcos^3 t)=(1/3)(∫(1-sin^2 t)dsint-tcos^3 t)=(1/3)(sint-1/3 sin^3 t-tcos^3 t)-C