Q) Integrate the following: ? (x^2+2)/(x^3+6x+1)^1/2 dx I have written the work

Question

Q) Integrate the following:

? (x^2+2)/(x^3+6x+1)^1/2 dx

I have written the work out, and I understand it up to here:

? (x^2+2)/(x^3+6x+1)^1/2 dx

Let u = x^3+6x+1

du/dx = 3x^2+6 which you can extract the common factor 3: 3(x^2+2)

Now:

? (x^2+2)/(u^1/2) * (du)/3(x^2+2)

(1/3)? (du)/(u^1/2)

Now I don't know what to do FROM HERE onwards. The next line confuses me, I don't know why she did this:

1/3 * u^(-1/2+1)/(-1/2+1) + C

and the ultimate answer is:

2/3* sqrt(u) + C which can be written as: 2/3 * sqrt(x^3+6x+1) +C

Please explain how did she get to that step from the part I was stuck on,

Explanation / Answer

The point where you are integrating 1/3 (du/sqrt(u)) integral of 1/3 u^-1/2 du (1/3) (2) u ^ 1/2 when you do the substitution for integration, you always have to plug back in the du and it gets simpler You still have to use the basic integral of something to an exponent integral of u^n du = U^(n+1)/(n+1)

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