stuck on integration problem?

Question

∫ [x^9 sqrt(x^5 - 3) ] dx

So this is what i did

∫ [x^9 sqrt(x^5 - 3) ] dx = ∫ [x^5 * x^4 * sqrt(x^5 - 3) ] dx

let u = x^5 then du = 5x^4 dx

∫ u sqrt(u - 3) du

Now what? stuck.... Do substition again?

Answer 1

Well you made a great awesome start...

Now lets let...

v = u-3

So dv = du

Our problem becomes

∫ (v+3)sqrt(v) dv

We can distribute the radical

∫ vsqrt(v) + 3sqrt(v) dv

∫ v^(3/2) + 3v^(1/2) dv

2/5 *v^(5/2) + 2*v^(3/2)

Back substitute

2/5(u-3)^(5/2) + 2(u-3)^(3/2)

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

There is your answer.

Answer 2

That's what I'd do.

v = u-3, dv = du gives you ∫(v+3)sqrt(v) dv, that is,
∫[v^(3/2) + 3v^(1/2)]dv

If instead you use u=x^5 - 3 in your initial substitution, you'll end up with the same thing (but avoiding the second substitution).

Answer 3

instead of letting u=x^5, include the -3 in it... so: u=x^5-3, du=5x^4 dx:
(1/5) ∫ u sqrt(u) du = ∫ u^(2/2) * u^(1/2) du
(1/5) ∫ u^(3/2) du <---- then use the power rule.. this you should know how to do now

Answer 4

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