Question

What is the antiderivative of `(x+1)(2x-1)`?

Answer 1

`int(x+1)(2x-1)dx=2/3x^3+1/2x^2-x+"c"`

Explanation:

Find an antiderivative of a function is the same as finding the integral.

So we want to find `int(x+1)(2x-1)dx=int2x^2+x-1dx`

To find this, we use the power rule

`intx^n dx=1/(n+1)x^(n+1)+"c"`

`rArrint2x^2+x-1dx=2/3x^3+1/2x^2-x+"c"`

Answer 2

`=>(2/3x^2+1/2x-1)x + C`

where `C` is an arbitrary constant from the integration.

Explanation:

We start with:

`\int(x+1)(2x-1)dx`

We multiply the two binomials:

`=\int(2x^2+2x-x-1)dx`

`=\int(2x^2+x-1)dx`

We break up the integral into three separate integrals:

`=\int2x^2dx + \intxdx + \int -dx`

`=color(blue)(2intx^2dx) + color(orange)(intxdx)-color(green)(intdx)`

Now we integrate:

`=color(blue)(2(1/3x^3))+color(orange)(1/2x^2)-color(green)(x)+C`

Simplifying:

`=2/3x^3+1/2x^2-x+C`

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

where `C` is an arbitrary constant from the integration.