Integration problems.. help urgent?

Question

Question improper integral

1) Integrate(O to Pi) 1/( Squareroot x + Sinx)
use the direct comparison test

2)integrate(o to 3) 1 / (xsquare-11x+24)

Question integration

3) integrate sin(2x). lncosxdx
use integration by part

Answer 1

1)
I think the first question only wants to know if the improper integral converges or not.

The improper integral in problem #1 does converge. Compare it with the convergent improper integral(0 to Pi) 1 / Squareroot x.

I don't think an antiderivative can be found for this one.

2)
The improper integral diverges.

The denominator of the integrand can be factored as
(x-3)(x-8). You can integrate this one using partial fractions. However, the upper limit is going to cause this integral to diverge.

3)
Using integration by parts and the trig. identity sin(2x)=2sin(x)cos(x).

Let
U = ln( cos(x) ) then dU = - sin(x) / cos(x)
dV=sin(2x)dx = 2 sin(x) cos(x) dx then V = -cos(x)^2

Thus
integral( sin(2x) ln(cos(x)) dx )
= - cos(x)^2 ln( cos(x) ) - integral( sin(x) cos(x) dx )
= - cos(x)^2 ln( cos(x) ) - 1/2 sin(x)^2 + C

Good Luck!

Answer 2

3♦ y= ∫ dx* sin(2x) *ln cos x;
u= ln cos x, du = -dx* tan(x);
dv = dx*sin(2x), v= -0.5*cos(2x);
♣ y= ln cos x * -0.5*cos(2x) - ∫ 0.5*cos(2x)* dx* tan(x) =
= -0.5*cos(2x)* ln cos x -0.5∫ (2sinx*cosx –tanx)* dx =
= -0.5*cos(2x)* ln cos x +0.25*cos(2x) -0.5* ln cos x +C;
♥ enuff 4 smart shyt like u!