Evaluate an Integral Using a Specific Hint

[LilTaru]

Homework Statement

\int dx/(x⁴ + 16)

Homework Equations

Hint: With a>0, x⁴ + a² = (x² + \sqrt{}2ax + a)(x² - \sqrt{}2ax + a)

The Attempt at a Solution

I've plugged this into the equation, which leaves me with:

\int dx/[(x² + \sqrt{}2ax + a)(x² - \sqrt{}2ax + a)]

But now I am sooooo confused! Where do I go from here? This hint does not seem so helpful... Please help!

 

[rock.freak667]

You could try splitting it into partial fractions now, which may make it easier.

 

[LilTaru]

How would I do that?

 

[rock.freak667]

Rewrite the 1/hint

as


\frac{1}{{(x^2+ \sqrt{2a}x + a}{( x^2 - \sqrt(2a) + a)}} = \frac{Ax+B}{(x^2+ \sqrt{2a}x + a} + \frac{Cx+D}{ x^2 - \sqrt(2a) + a}


then bring the right side to the same denominator and equate the numerators. Then choose values of x to get the constants A,B,C,D.

 

[LilTaru]

Yeah. This was my first approach but if you plug in the value of a = 4, it seems the two factors of the hints never equal 0... Maybe this is just me.

 

[Char. Limit]

Yeah. This was my first approach but if you plug in the value of a = 4, it seems the two factors of the hints never equal 0... Maybe this is just me.

If you can't pick values of x that decide certain values, there is another option, although it's not as fun: find the powers of x on each side and equate them to get a system of equations that you can solve.

 

[rock.freak667]

Yeah. This was my first approach but if you plug in the value of a = 4, it seems the two factors of the hints never equal 0... Maybe this is just me.

They don't need to equal to zero necessarily, the equation would be true for all values of x.

 

[LilTaru]

Okay... I tried to solve for the constants but I am stuck... I get this equation:

1 = (Ax + B)(x² - 2*sqrt(2)*x + 4) + (Cx + D)(x² + 2*sqrt(2)*x + 4)

I can't see how to solve this either by finding an x where the factors are zero or by solving a system of equations... How can I turn this into a system of equations?

 

[rock.freak667]

Okay... I tried to solve for the constants but I am stuck... I get this equation:

1 = (Ax + B)(x² - 2*sqrt(2)*x + 4) + (Cx + D)(x² + 2*sqrt(2)*x + 4)

I can't see how to solve this either by finding an x where the factors are zero or by solving a system of equations... How can I turn this into a system of equations?

Right, well you can multiply it out and equate coefficients OR you can put in values for x and get equations. For example, you can choose x=2,3,etc. and you will get equations to solve.

 

[LilTaru]

But I thought when you put in values of x it should make one of the factors be 0 so that you can solve for the other one?

 

[rock.freak667]

But I thought when you put in values of x it should make one of the factors be 0 so that you can solve for the other one?

I did not mean it like that.

You have 4 unknowns and the equation is true for all values of x.

So you can put in 4 different values for x and get 4 equations.

 

[LilTaru]

I've tried solving the system of equations but cannot seem to get it not matter how hard I try... can someone please help me solve the system? After that I know how to do the integral.

 

[rock.freak667]

What equations did you get?

 

[LilTaru]

Main equation: 1 = (Ax + B)(x² - 2\sqrt{}2*x + 4) + (Cx + D)(x² + 2\sqrt{}2*x + 4)

Set x = 0: 1/4 = B + D

Set x = sqrt(8): 1 = 8*sqrt(2)A + 8B + 40*sqrt(2)C + 20D

Set x = sqrt(2): 1 = -2*sqrt(2)A - 2B + 28*sqrt(2)C + 14D

Set x = 3*sqrt(2): 1 = 30*sqrt(2) + 10B + 102*sqrt(2) + 34D

Which by looking at the answers seems impossible plus possibly horribly wrong. Can someone tell me what they get please?!

 

[LilTaru]

After looking on the internet for a sample problem like this... I have found I am retarded... Figured out the problem: equated the coefficients wrong! Thank you all for your help! :P