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

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- Dec 20th 2009, 08:05 PMmellowdanosubstitution technique
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 - Dec 20th 2009, 08:33 PMoblixps
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. - Dec 21st 2009, 12:15 AMCaptainBlack
- Dec 21st 2009, 10:27 AMoblixps
whoops sorry i misread the question. but same idea.

- Dec 22nd 2009, 07:47 AMmellowdano
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))