The question i am given is:
Use a substitution technique and then the table of integration to integrate
∫ x^5 √(x^4 – 4) dx
Hint: x^5 = (x^3)(x^2)
Tips on how to start this solution would be appreciated.
Thanks
let u = x^4 - 4 and du = 4x^3 dx
x^5 can be split up into (x^3)(x^2) so when you make the substitutions, you integral will become: (1/4) integral of x^2 / sqrt(u) du. now to get x^2 in terms of u, take our original substitution u = x^4 - 4 and solve for x^2.
x^4 = u + 4 so x^2 = sqrt(u+4) so the integral will become: (1/4)integral of sqrt[(u+4)/(u)] du. the integrand can be further simplified to sqrt(1 + 4/x) = sqrt(1 + (2/sqrt(x))^2). now consult your integral table and find the form that matches this and finish up.
Would this be correct?
∫ x^5 √(x^4 – 4) dx
= ∫ ((x^2*x^2*x √(x^4 – 4)) dx
Let u = x^2, then du/2 = xdx
= ∫ 1/2 (u^2√u^2 – 4) du
Let a = 4 and a^2 = 2
= ∫(u^2√u^2 – a^2) du
= [x(u^2 – a^2)^3/2 / 4] + [a^2x√(u^2 – a^2) / 8] – (a^4 /8) ln(x + √(u^2 – a^2))