computation of integrals.

MathematicalPhysicist

i need to solve/prove the next two integrals:
[tex]\int\frac{dx}{u^2+u+4}[/tex]
and i need to show that:
[tex]\int_{0}^{\pi}\sqrt{1+sinx}dx=4[/tex]
the problem is that i have a clue to substitute u=sinx and then sin(pi)=0=sin0 so the integral should be equal zero, is it not?
ofcourse the integrand becomes: sqrt(1+u)/sqrt(1-u^2)

neutrino

For the first you can use the method of partial fractions.

You're right about the second. With the given limits, the integral is equal to zero.

Max Eilerson

substitute u = 1 + sin x. Are you sure the integral is equal to 4, not -4?

neutrino

But the plot (area) of the function sqrt(1+sin(x)) from 0 to pi seems to be non-zero!

Max Eilerson

It's non-zero.

MathematicalPhysicist

neutrino, how would i use partail fractions here?
i mean i need to decompose u^2+u+4 into a product of terms, but i have complex roots here.

Max Eilerson

complete the square.

MathematicalPhysicist

you mean something like this: u^2+u+4=(u-2)^2+5u
i still dont get an appropiate term to integrate.

neutrino

More like (u +0.5)^2 + 15/4

MathematicalPhysicist

ok, thanks.
btw, what about the second integral does it equal zero or it really does equal 4?

Max Eilerson

u^2+u+4= (u + 0.5)^2 + 15/4.

edit: too slow, the second integral should equal minus -4, I guess they're defining is it as area so you just need the modulus.

neutrino

Not -4. I just put the function through the Integrator and substituted the values, and I got 4. This graph is completely above the x-axis.

Max Eilerson

Btw, you will need to know what the derivative of the inverse tangent is.

Max Eilerson

Oops, I'm missing/added a minus somewhere. Didn't have the common senese to think about the graph :).

MathematicalPhysicist

wait a minute, then integral does converge to 4, care to explain how, where did i get it wrong?

dextercioby

For the integral try the sub

[tex] \tan\frac{x}{2}=t [/tex]

Daniel.

MathematicalPhysicist

but what's wrong with the substitution that im given a hint to use here?
i.e
sinx=u?