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干货！不定积分计算代数换元法应用举例六道例题
例题1：∫dx/[24+³√(32x+83)].
思路：变三次立方根无理式为有理式，变量替换t=³√(32x+83)。
解：设t=³√(32x+83)，则32x+83=t³，32dx=3t²dt;
∴∫dx/[24+³√(32x+83)]
=(1/32)*∫32dx/(24+t),
=(1/32)∫3*t²dt/(24+t),
=(3/32)∫[(t-24)(t+24)+24²]dt/(24+t),
=(3/32)*[∫(t-24)dt+24²∫dt/(t+24)],
=(3/32)*[1/2*t²-24t+24²*ln|t+24|+c],
=(3/64)*t-(9/8)t+(27/1)*ln|t+24|+C,
=(3/64)*³√(32x+83)²-(9/8)* ³√(32x+83)+(27/1)*ln|³√(32x+83)+24|+C.
例题2：∫[80+48(√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;
∴∫[80+48(√x)³]dx/(3+√x)，
=2∫(80+48 t³)tdt/(3+t)，
=2∫(48*t³-48*3*t²+48 *3²*t-1216)dt
+2*1216∫dt/(t+3),
=(1/2)*48*t⁴-(2/3)*48*3*t³+48*3²*
t²-2*1216t+2*1216*ln|t+3|+C,
=(1/2)*48*x²-(2/3)*48*3*√x³+48*3²*x-2*1216*√x
+2*1216*ln(√x+3)+C.
例题3：∫[√(67x+8)-17]dx/[17+√(67x+8)].
思路：变根式无理式为有理式，变量替换t=√(67x+8).
解：设t=√(67x+8)，则67x+8=t²，67dx=2tdt;
∴∫[√(67x+8)-17]dx/[17+√(67x+8)]
=(1/67)∫(t-17)*2tdt/(17+t),
=(1/67)[∫(2t-4*17)dt+(4/67)*17²∫dt/(t+17)],
=(1/67)(t²-4*17t)+(4/67)*17²*ln|t+17|+C,
=(1/67)( 67x+8-4*17*√(67x+8)+
(4/67)*17²*ln[(√(67x+8)+17)]+C,
例题4：∫x√(27-93x)dx.
思路：变根式无理式√(27-93x)为有理式，变量替换t=√(27-93x).
解：设t=√(27-93x)，则t²=27-93x,即：x=(1/93)(27-t²),
此时有：dx=-(1/93)*2tdt;
∴∫x√(27-93x)dx
=∫(1/93)(27-t²)*t*d[(1/93)(27-t²)]，
=∫(1/93)(27-t²)*t*[-(1/93)]*2tdt，
=-2*(1/93²)∫(27-t²)*t²dt,
=-2/93²*∫(27t²-t⁴)dt,
=-2/93²*[(2/3*27*t³-(1/5)*t⁵)]+C,
=-36/(1*93²)*t³+2/(5*93²)t⁵+C,
=-36/(1*93²)*√(27-93x)³+2/(5*93²)*√(27-93x)⁵+C,
例题5：∫(19ˣ*21ˣ)dx/(361ˣ-441ˣ).
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=(19/21)ˣ.
解：设t=(19/21)ˣ，则dt=t*ln(19/21)dx,
∫(19ˣ*21ˣ)dx/(361ˣ-441ˣ).
=∫(19/21)ˣdx/[1-(19/21)²ˣ],
=∫t*1/[t*ln(19/21)]dt/(1-t²),
=1/ln(21/19)*∫dt/(1-t²),
=1/(ln19-ln21)*[∫dt/(t-1)-∫dt/(t+1)],
=1/(2ln19-2ln21)*ln|(t-1)/(t+1)|+C,
=1/(2ln19-2ln21)*ln|[(19/21)ˣ-1]/[(19/21)ˣ+1]|+C,
=1/(2ln19-2ln21)*ln|(21ˣ-19ˣ)/(21ˣ+19ˣ)|+C.
例题6：∫dx/³√[(92x+91)²*(92x-91)⁴].
思路：代数式换元法，本题变量替换t=(92x+91)/(92x-91).
解：设t=(92x+91)/(92x-91)，有：
-2*92*91dx/(92x-91)²=dt，
即：dx=-(92x-91)²*dt/2*92*91.
代入积分函数有：∫dx/³√[(92x+91)²*(92x-91)⁴],
=∫dx/{³√[(92x+91)/(92x-91)]²*(92x-91)²] },
=[-1/(2*92*91)]∫(92x-91)²*dt/[³√t²*(92x-91)²] ,
=[-1/(2*92*91)]*∫dt/³√t² ,
=[-3/(2*92*91)]* ³√t+C,
=-3/(4186*2²)*³√t+C,
=-3/(4186*2²)*³√[(92x+91)/(92x-91)]+C。
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