Question

How do you find the antiderivative of `f(x)=1/4x^4-2/3x^2+4`?

Answer 1

`intf(x)dx=int(1/4x^4-2/3x^2+4)dx`

`=1/4intx^4dx-2/3intx^2dx+4intx^0dx`

`=1/4{x^(4+1)/(4+1)}-2/3{x^(2+1)/(2+1)}+4{x^(0+1)/(0+1)}`

`=1/20x^5-2/9x^3+4x+C,`

`because, intx^ndx=x^(n+1)/(n+1)+c.`

Enjoy Maths.!

Answer 2

`1/20x^5-2/9x^3+4x+c`

Explanation:

Integrate each term using the `color(blue)"power rule for integration"`

`color(orange)"Reminder " color(red)(bar(ul(|color(white)(2/2)color(black)(int(ax^n)=a/(n+1)x^(n+1))color(white)(2/2)|)))`

`rArrint(1/4x^4-2/3x^2+4)dx`

`=(1/4)/5x^5-(2/3)/3x^3+4x+c`

`=1/20x^5-2/9x^3+4x+c`

where c is the constant of integration.