Tackling Partial Fractions: What's Going on with the Numerator?

[canucks81]

Homework Statement

Use integration by parts to evaluate the integral

∫(7-6x) / (x²-4x+13)

The Attempt at a Solution

This is a question from my notes so I already have the solution but I'm not sure what's going on at this one specific step.

∫(7-6x) / (x²-4x+13)

= -∫(6x-7) / (x²-4x+13)

= -∫( 3(2x-4) + 5) / (x²-4x+13)

So I know 2x-4 is the derivative of the denominator but I don't know why the numerator is changing to 3(2x-4) + 5.

 

[Mark44]

Homework Statement

Use integration by parts to evaluate the integral

∫(7-6x) / (x²-4x+13)

The Attempt at a Solution

This is a question from my notes so I already have the solution but I'm not sure what's going on at this one specific step.

∫(7-6x) / (x²-4x+13)

= -∫(6x-7) / (x²-4x+13)

= -∫( 3(2x-4) + 5) / (x²-4x+13)

So I know 2x-4 is the derivative of the denominator but I don't know why the numerator is changing to 3(2x-4) + 5.

Please don't use the SIZE tags. What you wrote is legible enough without making it larger.

Ignoring the integration part for the moment, here's the algebra. What they're doing is writing the original fraction as
$$\frac{-3(2x - 4) - 5}{x^2 - 4x + 13} = \frac{-3(2x - 4)}{x^2 - 4x + 13} + \frac{-5}{x^2 - 4x + 13}$$

Now as an integration problem, the first fraction can be integrated using an ordinary substitution and the second can be done using a formula you've probably already seen.

Your title is misleading - I don't see this as a partial fractions problem. I also don't see this as in integration by parts problem, either.

 

[ehild]

Homework Statement

Use integration by parts to evaluate the integral

∫(7-6x) / (x²-4x+13)

The Attempt at a Solution

This is a question from my notes so I already have the solution but I'm not sure what's going on at this one specific step.

∫(7-6x) / (x²-4x+13)

= -∫(6x-7) / (x²-4x+13)

= -∫( 3(2x-4) + 5) / (x²-4x+13)

So I know 2x-4 is the derivative of the denominator but I don't know why the numerator is changing to 3(2x-4) + 5.

The numerator changes to -3[(2x-4) + 5], which is equal to 7-6x.

ehild