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    Solve and Explain Three Integrals

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    ∫ (x^5)/[(1+x^3)^(3/2)] dx
    ∫ {square root of [(x+3)/(x+1)]}dx
    ∫ (cot^3 v)[(csc v)^(3/2)] dx

    (I will use the $ sign for the integral sign)

    Problem #1:
    $ (x^5)/[(1+x^3)^(3/2)] dx

    the power 3/2 in the denominator is throwing me off greatly, as is
    the greater power (x^5) in the numerator.

    attempt 1: let u=x^2 (so u^(1/2)=x) and du=2xdx the integral becomes
    1/2$ [(u^2*u^(1/2))/(1+u*u^(1/2))^(3/2)]du
    = 1/2$ [(u^3/2)/(1+u^(3/2))^(3/2)]du

    at this point I wasn't sure of where to go so I restarted the
    problem using intergration by parts

    attempt 2: let u=1/[(1+x^3)^3] and du= {-9x^2/[(1+x^3)^4]dx let dv=
    (x^5)dx and v= 1/6(x^6)
    the integral was then
    =1/6(x^6)*[1/(x+1)^3] - $1/6(x^6)*(-9x^2/[(1+x^3)^4])dx
    =(x^6)/[6(x+3)^3]) - (9/6)$ x^2/[(x+3)^4]dx
    but I couldn't figure what to do with the x^2


    $ {square root of [(x+3)/(x+1)]}dx

    I can't get around the square root for the entire problem. I can't
    figure out the correct substitution method.

    At first I tried to ignore the squareroot and integrate by part like
    let u=x+3 and du=dx so dv=[1/(x+1)]dx and v=ln(x+1) but it didn't
    seem to work correctly.
    I thought maybe let u=(x+3)/(x+1) so the $u^(1/2) but the du
    wouldn't work out correctly. I thought of partial fraction
    integration but didn't know how to do that with the squareroot.

    $ (cot^3 v)[(csc v)^(3/2)] dx

    Once again, the fraction exponent is throwing me off. I know there
    are substitutions and spliting of integral, but I can't get rid of
    the (csc v)^(3/2)

    ${(cot^3 v)[(csc v)^(3/2)]}dv
    ${(cot^2 v)(cot v)[(csc v)^(3/2)]} dv cot^2 v= csc^2 v -1
    ${(csc^2 v -1)(cot v)[(csc v)^(3/2)]} dv
    ${(csc^2 v)(cot v)[(csc v)^(3/2)]} dv - ${(cot v)[(csc v)^(3/2)]}dv
    I know that I can use trig sub. for the csc^2 v and cot, but I don't
    know how to get rid of the csc v^(3/2) or even how to just get it to
    be a csc v.

    Any information that could help me with any of these three problems
    would be extremely appreciated.

    © BrainMass Inc. brainmass.com March 4, 2021, 6:25 pm ad1c9bdddf

    Solution Preview


    Solution of three calculus problems is attached.

    I have give file names in this way ...

    Solution Summary

    Three integrals are solved and explained. The solution is detailed and well presented. The response received a rating of "5" from the student who originally posted the question.