Integration Help - Which Method?

[BOAS]

Hello,

i'm having some trouble identifying the correct method to approach a problem. I don't have any working to show, so i'll explain why I think the methods I've considered are not correct.

Homework Statement

Integrate with respect to [itex]x[/itex] the following functions.

[itex]\int \frac{x}{(3-x)^{7}} dx[/itex]

Homework Equations

The Attempt at a Solution

I think the method of changing the variable is not suitable here because I can't see a link between the derivative of the denominator and the numerator.

Similarly I don't think it is possible to use the method that identifies the numerator as the derivative of the denominator allowing you to say the integral is equal to ln|f(x)| + c

I'd really appreciate a nudge in the right direction!

Thank you,

BOAS

 

[ehild]

Use the substitution u=3-x.

ehild

 

[BOAS]

Use the substitution u=3-x.

ehild

I'm struggling to make this work.

[itex]\int \frac{x}{(3-x)^{7}} dx[/itex]

Let [itex] u \equiv (3 - x)[/itex]

[itex]\frac{du}{dx} = -1[/itex]

[itex] ... du \equiv ... -1dx[/itex]

[itex]\int \frac{x}{(3-x)^{7}} dx = -x \int \frac{du}{u^{7}}[/itex]

(that seems wrong to me, I don't think we've ever been told it's ok to bring an x outside the integral sign)

I get;

[itex]\int \frac{x}{(3-x)^{7}} dx = \frac{x}{6(3-x)^6} + c[/itex]

 

[phyzguy]

You can't just pull out the x like you did. You have to replace the x in the numerator with x=3-u and leave it under the integral sign.

 

[Tanya Sharma]

I'm struggling to make this work.

[itex]\int \frac{x}{(3-x)^{7}} dx[/itex]

Let [itex] u \equiv (3 - x)[/itex]

[itex]\frac{du}{dx} = -1[/itex]

[itex] ... du \equiv ... -1dx[/itex]

[itex]\int \frac{x}{(3-x)^{7}} dx = -x \int \frac{du}{u^{7}}[/itex]

The item in red is wrong . You need to express 'x' present in the Nr also in terms of 'u' .After the substitution ,you should have ##\int \frac{u-3}{u^7} du##

 

[Ray Vickson]

Hello,

i'm having some trouble identifying the correct method to approach a problem. I don't have any working to show, so i'll explain why I think the methods I've considered are not correct.

Homework Statement

Integrate with respect to [itex]x[/itex] the following functions.

[itex]\int \frac{x}{(3-x)^{7}} dx[/itex]

Homework Equations

The Attempt at a Solution

I think the method of changing the variable is not suitable here because I can't see a link between the derivative of the denominator and the numerator.

Similarly I don't think it is possible to use the method that identifies the numerator as the derivative of the denominator allowing you to say the integral is equal to ln|f(x)| + c

I'd really appreciate a nudge in the right direction!

Thank you,

BOAS

Others have suggested a change of variables, but if you do straight integration by parts you can bypass that: just set ##u = x, \: dv = dx/(3-x)^7 = (3-x)^{-7} \, dx##.

 

[statdad]

Think again about your substitution: if [itex] u = 3 -x [/itex] then [itex] x = 3 - u [/itex]. How will that help your integrand