ज्ञात करें `int x^5/sqrt(1+x^3)dx`

Explanation

यहां `x^5/sqrt(1+x^2)=x^3/sqrt(1+x^3)x^2` तथा `x^3/sqrt(1+x^3)=(1+x^3-1)/sqrt(1+x^3)(1+x^2)` का एक फलन है तथा `(1+x^3)` का `d.c.=3x^2=3`. इसीलिए `z=1+x^3` रखे |
अब, `int x^5/sqrt(1+x^3)dx=intx^3/sqrt(1+x^3)x^2dx=int((z-1)/sqrtz)(dz)/3`
`=1/3int (z-1)/sqrtzdz=1/3int (z/sqrtz-1/sqrtz)dz=1/3int(sqrtz-z^(-1/2))dz`
`=1/3[z^(3/2)/(3/2)-z^(1/2)/(1/2)]+c=1/3(2/3z^(3/2)-2sqrtz)+c`
`=2/9sqrtz(z-3)+c=2/3sqrt(1+x^3)(x+x^3-3)+c=2/9sqrt(1+x^3)(x^3-2)+c`