Question

integrate:

(x^2-3)sqrt (x+5)

Answer 1

Let u^2 = x+5

2u du =dx

x = u^2 - 5
x^2 - 3 = u^4 - 10u^2 + 22

Now the integrand becomes
(u^4 - 10u^2 + 22)(u)(2u du)
= 2u^6 - 20u^5 + 44u^2 du

That is easy to integrate. Substituting back to x at the end you have

2/21 (3x^2 - 12x + 19) (x+5)^(3/2) + C