例题1：∫dx/[3+3√(85x+38)].
思路：变三次立方根无理式为有理式，变量替换t=3√(85x+38)。
解：设t=3√(85x+38)，则85x+38=t3，85dx=3t2dt;
∴∫dx/[3+3√(85x+38)]
=(1/85)*∫85dx/(3+t),
=(1/85)∫3*t2dt/(3+t),
=(3/85)∫[(t-3)(t+3)+32]dt/(3+t),
=(3/85)*[∫(t-3)dt+32∫dt/(t+3)],
=(3/85)*[1/2*t2-3t+32*ln|t+3|+c],
=(3/170)*t-(9/170)t+(27/170)*ln|t+3|+C,
=(3/170)*3√(85x+38)2-(9/170)* 3√(85x+38)+(27/170)*ln|3√(85x+38)+3|+C.
例题2：∫[95+24(√x)3]dx/(3+√x).
思路：变平方根无理式为有理式，变量替换t=√x.
解：设t=√x，则x=t2，dx=2tdt;
∴∫[95+24(√x)3]dx/(3+√x)，
=2∫(95+24 t3)tdt/(3+t)，
=2∫(24*t3-24*3*t2+24 *32*t-553)dt
+2*553∫dt/(t+3),
=(1/2)*24*t?-(2/3)*24*3*t3+24*32*
t2-2*553t+2*553*ln|t+3|+C,
=(1/2)*24*x2-(2/3)*24*3*√x3+24*32*x-2*553*√x
+2*553*ln(√x+3)+C.
例题3：∫[√(66x+7)-23]dx/[23+√(66x+7)].
思路：变根式无理式为有理式，变量替换t=√(66x+7).
解：设t=√(66x+7)，则66x+7=t2，66dx=2tdt;
∴∫[√(66x+7)-23]dx/[23+√(66x+7)]
=(1/66)∫(t-23)*2tdt/(23+t),
=(1/66)[∫(2t-4*23)dt+(2/33)*232∫dt/(t+23)],
=(1/66)(t2-4*23t)+(2/33)*232*ln|t+23|+C,
=(1/66)( 66x+7-4*23*√(66x+7)+
(2/33)*232*ln[(√(66x+7)+23)]+C。
例题4：∫x√(79-21x)dx.
思路：变根式无理式√(79-21x)为有理式，变量替换t=√(79-21x).
解：设t=√(79-21x)，则t2=79-21x,即：x=(1/21)(79-t2),
此时有：dx=-(1/21)*2tdt;
∴∫x√(79-21x)dx
=∫(1/21)(79-t2)*t*d[(1/21)(79-t2)]，
=∫(1/21)(79-t2)*t*[-(1/21)]*2tdt，
=-2*(1/212)∫(79-t2)*t2dt,
=-2/212*∫(79t2-t?)dt,
=-2/212*[(2/3*79*t3-(1/5)*t?)]+C,
=-316/(3*212)*t3+2/(5*212)t?+C,
=-316/(3*212)*√(79-21x)3+2/(5*212)*√(79-21x)?+C。
例题5：∫(14?*17?)dx/(196?-289?).
思路：将被积函数进行变形，再进行变量替换，本题变量替换t=(14/17)?.
解：设t=(14/17)?，则dt=t*ln(14/17)dx,
∫(14?*17?)dx/(196?-289?).
=∫(14/17)?dx/[1-(14/17)2?],
=∫t*1/[t*ln(14/17)]dt/(1-t2),
=1/ln(17/14)*∫dt/(1-t2),
=1/(ln14-ln17)*[∫dt/(t-1)-∫dt/(t+1)],
=1/(2ln14-2ln17)*ln|(t-1)/(t+1)|+C,
=1/(2ln14-2ln17)*ln|[(14/17)?-1]/[(14/17)?+1]|+C,
=1/(2ln14-2ln17)*ln|(17?-14?)/(17?+14?)|+C.
例题6：∫dx/3√[(77x+73)2*(77x-73)?].
思路：代数式换元法，本题变量替换t=(77x+73)/(77x-73).
解：设t=(77x+73)/(77x-73)，有：
-2*77*73dx/(77x-73)2=dt，
即：dx=-(77x-73)2*dt/2*77*73.
代入积分函数有：∫dx/3√[(77x+73)2*(77x-73)?],
=∫dx/{3√[(77x+73)/(77x-73)]2*(77x-73)2] },
=[-1/(2*77*73)]∫(77x-73)2*dt/[3√t2*(77x-73)2] ,
=[-1/(2*77*73)]*∫dt/3√t2 ,
=[-3/(2*77*73)]* 3√t+C,
=-3/(11242*12)*3√t+C,
=-3/(11242*12)*3√[(77x+73)/(77x-73)]+C。
