Question

What is the antiderivative of `(2x^2 - 5)(x^2 + 1)`?

Answer 1

`(2x^5)/5-x^3-5x+C`

Explanation:

The question you are asking can be written as:

`int(2x^2-5)(x^2+1)dx`

First, simplify the integral by distributing the binomials.

`=int(2x^4-3x^2-5)dx`

Split this up into three separate integrals:

`=2intx^4dx-3intx^2dx-5intdx`

For the first two integrals, use the rule:`" "intx^n=x^(n+1)/(n+1)+C`

For the third, use:`" "intdx=x+C`

The multiplicative constants attached to the fronts of the integrals will just be multiplied with the rest of the antiderivatives:

Applying the rules gives:

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

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