How to determine this integral? Thank you

[williamwong0402]

Homework Statement

Homework Equations

k∫[ƒ(x)]ⁿ ƒ^'(x) dx

The Attempt at a Solution

i tried to using algebraic substitution to determine that i had let u = 1-x or X²-2x+1 or x or root(x) but it still cannot solve it.
Please give me hint how to solve it.
Thank you
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[Jonathan Scott]

I'd just expand it as a sum of the three terms, each of which is a half-integer power of x, integrate each separately and then simplify the result sum.

 

[eys_physics]

Expand $(1-x)^2=1-2x+x^2$ and then treat each term separately.

 

[PeroK]

The substitution $u = \sqrt{x}$ should have simplified it. Perhaps you ought to post what you did to see where you went wrong.

 

[williamwong0402]

$$ ∫ \frac{ (1-x)^2 }{ 2 \sqrt{x} } dx$$
$$ =∫ \frac{ (x^2+2x+1) }{ 2 \sqrt{x} } dx$$
$$ Let u = \sqrt{x} $$
$$ dx= \frac{ 2du }{ x^{\frac{ -1 }{ 2 }} } $$
$$ ∫\frac{ (x^2+2x+1) }{ 2u } \frac{ 2du }{ x^{\frac{ -1 }{ 2 }} }$$

and then i have no idea what is next step
thank you very much

 

[PeroK]

$$ ∫ \frac{ (1-x)^2 }{ 2 \sqrt{x} } dx$$
$$ =∫ \frac{ (x^2+2x+1) }{ 2 \sqrt{x} } dx$$
$$ Let u = \sqrt{x} $$
$$ dx= \frac{ 2du }{ x^{\frac{ -1 }{ 2 }} } $$
$$ ∫\frac{ (x^2+2x+1) }{ 2u } \frac{ 2du }{ x^{\frac{ -1 }{ 2 }} }$$

and then i have no idea what is next step
thank you very much

The point of a substitution is to replace one variable with another, not to mix the two. You need to replace all the terms in $x$ with the relevant term in $u$.

 

[Ray Vickson]

$$ ∫ \frac{ (1-x)^2 }{ 2 \sqrt{x} } dx$$
$$ =∫ \frac{ (x^2+2x+1) }{ 2 \sqrt{x} } dx$$
$$ Let u = \sqrt{x} $$

and then i have no idea what is next step
thank you very much

Using any kind of substitution in this problem is unhelpful; it just makes the problem harder, not easier! Just integrate the three terms separately.

 

[williamwong0402]

thank you ~
The answer:
$$ \frac{ x \frac{ 5 }{ 2 } }{ 5 } + \frac{ 2x \frac{ 3 }{ 2 } }{ 3 } + \sqrt{x} $$
Am i right ?

 

[PeroK]

Using any kind of substitution in this problem is unhelpful; it just makes the problem harder, not easier! Just integrate the three terms separately.

If the OP doesn't understand how to do substitution, it's as well to get that fixed. He'll need to get to grips with it sooner or later!

 

[Jonathan Scott]

thank you ~
The answer:
$$ \frac{ x \frac{ 5 }{ 2 } }{ 5 } + \frac{ 2x \frac{ 3 }{ 2 } }{ 3 } + \sqrt{x} $$
Am i right ?

You lost a minus sign when you expanded the square, and your formatting is a bit scrambled.

 

[PeroK]

thank you ~
The answer:
$$ \frac{ x \frac{ 5 }{ 2 } }{ 5 } + \frac{ 2x \frac{ 3 }{ 2 } }{ 3 } + \sqrt{x} $$
Am i right ?

You should never (or rarely) have to ask whether an indefinite integral answer is correct. You should differentiate your answer and see whether you get the original function back. You have, however, forgotten the constant of integration.

 

[williamwong0402]

ok ~ thank you so much