Question

Find the antiderivative for the function f(x) = (x – 11)(x + 13). Please help

Answer 1

Expanding the product is simple enough:

$(x-11)(x+13)=x^2+2x-143$

Then

$\displaystyle\int(x-11)(x+13)\,\mathrm dx=\dfrac{x^3}3+x^2-143x+C$

We could also use a subsitution like $u=x-11$, so that $u+24=x+13$ and $\mathrm du=\mathrm dx$:

$\displaystyle\int u(u+24)\,\mathrm du=\int(u^2+24u)\,\mathrm du=\dfrac{u^3}3+12u^2+C$

$\implies\displaystye\int(x-11)(x+13)\,\mathrm dx=\dfrac{(x-11)^3}3+12(x-11)^2+C$

(which is differs from the first result by a constant, so it's still valid)