1. ## Trigonometric integral....

any help with this?

$\displaystyle \int^{\pi/3}_{0} \frac{\sin^3x}{\sqrt{\cos x}} \, dx$

I first divided the $\displaystyle \sin^3 x$ into two: $\displaystyle \frac{\sin^2 x \sin x}{ \sqrt{\cos x}}$

then got $\displaystyle \frac{\sin^2 x \sin x}{ \cos^{1/2} x}$

I get confused here but I think it is....

(-cos^2x) sinx cos^1/2x

i get stuck here....

2. Originally Posted by dude487
any help with this?

$\displaystyle \int\frac{sin^3x}{cosx}$
Hello,

I fixed the latex. should be { } not ( )

have you tried substituting $\displaystyle t=\sin(x)$ ?
$\displaystyle dt=\cos(x) ~dx$

so we have $\displaystyle \int\frac{t^3}{\cos^2(x)} ~dt$
and $\displaystyle \cos^2(x)=1-\sin^2(x)=1-t^2$

which will give a partial decomposition...

so let's try $\displaystyle u=\cos(x)$
$\displaystyle du=-\sin(x) ~dx$

so we have $\displaystyle \int \frac{-\sin^2(x)}{u} ~du=\int\frac{u^2-1}{u} ~du=\int u-\frac 1u ~du$

much better ! can you finish it ?

--------------------------------------------------------
okay, you changed the problem... however, it doesn't change the method. the best substitution seems to be u=cos(x)
as for the boundaries, i'll leave it to you =)

3. Originally Posted by dude487
any help with this?

$\displaystyle \int^{\pi/3}_{0} \frac{\sin^3x}{\sqrt{\cos x}} \, dx$

I first divided the $\displaystyle \sin^3 x$ into two: $\displaystyle \frac{\sin^2 x \sin x}{ \sqrt{\cos x}}$

then got $\displaystyle \frac{\sin^2 x \sin x}{ \cos^{1/2} x}$

I get confused here but I think it is....

(-cos^2x) sinx cos^1/2x

i get stuck here....
$\displaystyle \int_0^{\pi /3} {\frac{{{{\sin }^3}x}}{{\sqrt {\cos x} }}dx} = \int_0^{\pi /3} {\frac{{\sin x{{\sin }^2}x}}{{\sqrt {\cos x} }}dx} = \int_0^{\pi /3} {\frac{{\sin x\left( {1 - {{\cos }^2}x} \right)}}{{\sqrt {\cos x} }}dx} =$

$\displaystyle = \int_0^{\pi /3} {\sin x\left( {\frac{1}{{\sqrt {\cos x} }} - \sqrt {{{\cos }^3}x} } \right)dx} = \int_0^{\pi /3} {\left( {\sqrt {{{\cos }^3}x} - \frac{1}{{\sqrt {\cos x} }}} \right)d\left( {\cos x} \right)} =$

$\displaystyle = \left. {\left( {\frac{2}{5}\sqrt {{{\cos }^5}x} - 2\sqrt {\cos x} } \right)} \right|_0^{\pi /3} = \left. {2\sqrt {\cos x} \left( {\frac{1} {5}{{\cos }^2}x - 1} \right)} \right|_0^{\pi /3} =$

$\displaystyle = 2 \cdot \frac{1}{{\sqrt 2 }}\left( {\frac{1}{5} \cdot \frac{1}{4} - 1} \right) - 2\left( {\frac{1}{5} - 1} \right) = \sqrt 2 \left( {\frac{1} {{20}} - 1} \right) - 2\left( {\frac{1}{5} - 1} \right) = \frac{8} {5} - \frac{{19\sqrt 2 }}{{20}}.$