Definite integral involving partial fractions

[Obliv]

Homework Statement

Homework Equations

trigonometric identities

The Attempt at a Solution

I did a trig substitution of u=tan(θ/2) and from that I could substitute cos(θ) = 1-u²/1+u²

dθ = 2/(1+u²)
du = 1/2 sec²(θ/2) dθ

I simplified a bit and changed the bounds to get 2du/(5u² + 1)(1 + u²)² with lower bound 0 and upper bound 1.

I think at this point I have to do partial fractions. Do I need 3 linear terms or 2? I tried Ax+B/5u²+1 + Cx + D/(1+u²)² + Ex+F/(1+u²) = 2/(5u² + 1)(1 + u²)

I have no idea how to evaluate such a complicated system of equations. I could really use some guidance here.

 

[andrewkirk]

First you need to fix up the errors.

You have written x in the numerators where you should have written u.
You have omitted parentheses around your numerators and denominators.
You have omitted the exponent 2 in the second factor of the denominator on the RHS.

Once you've done that, multiply both sides by the RHS denominator and you'll have a polynomial equation of order 6 with six unknown constants. Put it all on one side and then you get seven equations by equating the coefficient of each power of u to zero.

Those equations will enable you to solve to find the constants.

 

[SammyS]

Homework Statement

Homework Equations

trigonometric identities

The Attempt at a Solution

I did a trig substitution of u=tan(θ/2) and from that I could substitute cos(θ) = (1-u²)/(1+u²)

Throw some parentheses in where needed, and that looks right.

dθ = 2/(1+u²)

There should be a du on the right hand side of the above.
$\displaystyle \ d\theta=\frac{2}{1+u^2}du \$

I don't see any need to use the following.

du = 1/2 sec²(θ/2) dθ

I simplified a bit and changed the bounds to get 2du/(5u² + 1)(1 + u²)² with lower bound 0 and upper bound 1.

Check your Algebra. I get that the (1+ u²) should cancel, not be squared.

I think at this point I have to do partial fractions. Do I need 3 linear terms or 2? I tried Ax+B/5u²+1 + Cx + D/(1+u²)² + Ex+F/(1+u²) = 2/(5u² + 1)(1 + u²)

I have no idea how to evaluate such a complicated system of equations. I could really use some guidance here.

 

[Obliv]

I don't see any need to use the following.

Check your Algieba. I get that the (1+ u²) should cancel, not be squared.

thanks a ton. I didn't catch that at first. I actually re-checked my algebra like 10 times and kept getting different answers. It's sad to see gaps in my knowledge but I guess this gives me good reason to strengthen those areas.