Question

Question #4f6e6

Answer 1

`int(x^6-1)/(1+x^2)dx=1/5x^5-1/3x^3+x-2arctan(x)+C`

Explanation:

Split up the integral:

`I=int(x^6-1)/(1+x^2)dx=intx^6/(1+x^2)dx-intdx/(1+x^2)`

The second is the arctangent integral:

`I=intx^6/(1+x^2)dx-arctan(x)`

Perform long division on what remains, or write the numerator as follows for the same result:

`I=int(x^4(1+x^2)-x^2(1+x^2)+(x^2+1)-1)/(1+x^2)dx-arctan(x)`

Dividing:

`I=intx^4dx-intx^2dx+intdx-intdx/(1+x^2)-arctan(x)`

`I=1/5x^5-1/3x^3+x-2arctan(x)+C`