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

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Answer 1 · 2016-03-26T18:26:57

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

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:

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

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

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