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*t⁴-(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-t⁴)dt,
=-2/21²*[(2/3*79*t³-(1/5)*t⁵)]+C,
=-316/(3*21²)*t³+2/(5*21²)t⁵+C,
=-316/(3*21²)*√(79-21x)³+2/(5*21²)*√(79-21x)⁵+C。
例题5：∫(14ˣ*17ˣ)dx/(196ˣ-289ˣ).
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)ˣ.
解：设t=(14/17)ˣ，则dt=t*ln(14/17)dx,
∫(14ˣ*17ˣ)dx/(196ˣ-289ˣ).
=∫(14/17)ˣdx/[1-(14/17)²ˣ],
=∫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)ˣ-1]/[(14/17)ˣ+1]|+C,
=1/(2ln14-2ln17)*ln|(17ˣ-14ˣ)/(17ˣ+14ˣ)|+C.
例题6：∫dx/³√[(77x+73)²*(77x-73)⁴].
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)⁴],
=∫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。
