Mth 174
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I'm having trouble calculating integrals that are in the form of a fraction and that typically have a square root in the denominator.
integral(y^2 / (sqrt(y+9)
let w= y+9 and then set y= w-9, rewrite in terms of w and get w^2-18w+81 * 2w^1/2 = 2w- 126w^1/2, antiderivative is u^2 + 84w^ 3/2, then replace w with y+9. Is this correct? Im having difficulty with fraction integrals mostly involving substitution.
If w = y + 9 then dw = dy so the integrand (y^2 / sqrt(y + 9)) dy becomes
((w – 9)^2 / sqrt(w) ) dw
= (w^2-18w+81)/sqrt(w) dw
= (w^(3/2) – 18 w^(1/2) + 81 w^(-1/2) )dw,
which can be integrated term-by-term to give you 2/5 w^(5/2) – 12 w^(3/2) + 162 w^(1/2) + c.
Then substitute w = 9 - y to obtain your final result.
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