Can anyone please help me with a u substitution integration question: Work out the integral: `int x/(a+bx)^{3/2}dx`   let `u=a+bx`  Please help if you can very much appreciated if you do

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In this case, since the substitution is given in the question, we just need to carry out the procedure for substitution.

Since `u=a+bx` , then `du=bdx` so `dx=1/bdu` and `x=1/b(u-a)` .  This means the integral becomes:

`int x/(a+bx)^{3/2}dx`

`=1/b^2int {u-a}/u^{3/2}du`   now split the integral into two integrals

`=1/b^2int u/u^{3/2}du-a/b^2int 1/u^{3/2}du`   simplify...

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In this case, since the substitution is given in the question, we just need to carry out the procedure for substitution.

Since `u=a+bx` , then `du=bdx` so `dx=1/bdu` and `x=1/b(u-a)` .  This means the integral becomes:

`int x/(a+bx)^{3/2}dx`

`=1/b^2int {u-a}/u^{3/2}du`   now split the integral into two integrals

`=1/b^2int u/u^{3/2}du-a/b^2int 1/u^{3/2}du`   simplify the integrands

`=1/b^2intu^{-1/2}du-a/b^2int u^{-3/2}du`   now use power law

`=2/b^2u^{1/2}+{2a}/b^2u^{-1/2}+C`  where C is the constant of integration

`={2(u+a)}/{b^2u^{1/2}}+C`

The integral evaluates to `{2(u+a)}/{b^2u^{1/2}}+C` .

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