Let I=∫dxx2(1+x4)34
=∫dxx2[x4(1x4+1)]34
=∫dx(x4)34(1x4+1)34
=∫dxx2+3(1x4+1)34
=∫dxx5(1x4+1)34
Let t=1x4+1⇒dt=−4x5dx⇒−dt4=dxx5
I=∫−dt4t34
=−14∫−dtt34
=−14∫t−34dt
=−14⎡⎢
⎢
⎢⎣t−34+1−34+1⎤⎥
⎥
⎥⎦
=−14t1414
=−t14+c where c is the constant of integration
∴I=−(1x4+1)14+c where t=\dfrac{1}{{x}^{4}}+1$