Reduction Formulae Question : In= ∫x(cos^n(x))

[k1902]

Homework Statement

Let In = ∫x(cos^n(x)) with limits between x=π/2, x=0 for n≥0
i) Show that nIn=(n-1)In-2 -n^-1 for n≥2
ii) Find the exact value of I3

Homework Equations

∫u'v = uv-∫uv' is what I use for these questions

The Attempt at a Solution

Rewritten as ∫ xcos^n-1(x) cosx
u'=cosx v= xcos^n-1x
u= sinx v'= cos^n-1 -x(n-1)sinxcos^n-2x
But I can't seem to write it in the form it asks for.

 

[LCKurtz]

Homework Statement

Let In = ∫x(cos^n(x)) with limits between x=π/2, x=0 for n≥0
i) Show that nIn=(n-1)In-2 -n^-1 for n≥2
ii) Find the exact value of I3

Your questions are indecipherable. Strange notation and no parentheses.

 

[Mark44]

Homework Statement

Let In[ i] = ∫x(cos^n(x)) with limits between x=π/2, x=0 for n≥0
i) Show that nIn=(n-1)In-2 -n^-1 for n≥2
ii) Find the exact value of I3

Part of the problem is that if you type i in brackets, the browser things you mean the change the font type to italics. Besides that, it's hard to tell exactly what the problem is you're trying to solve.

Here's what I think you meant.
Let $I_n = \int_0^{\pi/2} x \cos^n(x) dx$, with $n \ge 0$
i) Show that $n I_n = (n - 1)I_{n-2} - n^{n - 1}$, for $n \ge 2$.
ii) Find the exact value of $I_3$.
Is this anywhere close to what you're asking?

PS - I used LaTeX to format what I wrote. We have a tutorial here: https://www.physicsforums.com/help/latexhelp/. This is under the INFO menu, under Help/How-to.

Homework Equations

∫u'v = uv-∫uv' is what I use for these questions

The Attempt at a Solution

Rewritten as ∫ xcos^n-1(x) cosx
u'=cosx v= xcos^n-1x
u= sinx v'= cos^n-1 -x(n-1)sinxcos^n-2x
But I can't seem to write it in the form it asks for.

 

[k1902]

Part of the problem is that if you type i in brackets, the browser things you mean the change the font type to italics. Besides that, it's hard to tell exactly what the problem is you're trying to solve.

Here's what I think you meant.
Let $I_n = \int_0^{\pi/2} x \cos^n(x) dx$, with $n \ge 0$
i) Show that $n I_n = (n - 1)I_{n-2} - n^{n - 1}$, for $n \ge 2$.
ii) Find the exact value of $I_3$.
Is this anywhere close to what you're asking?

PS - I used LaTeX to format what I wrote. We have a tutorial here: https://www.physicsforums.com/help/latexhelp/. This is under the INFO menu, under Help/How-to.

Sorry, I'm new here and have no idea how to format the text.
But yes that's what I was trying to write , except part i) is i) Show that $n I_n = (n - 1)I_{n-2} - n^{- 1}$, for $n \ge 2$.
Thanks for the help with formatting :)

 

[k1902]

Your questions are indecipherable. Strange notation and no parentheses.

Apologies for that, here's the cleaned up, comprehensible version of the question (thanks to Mark44):
Let $I_n = \int_0^{\pi/2} x \cos^n(x) dx$, with $n \ge 0$
i) Show that $n I_n = (n - 1)I_{n-2} - n^{- 1}$, for $n \ge 2$.
ii) Find the exact value of $I_3$.

 

[Mark44]

Apologies for that, here's the cleaned up, comprehensible version of the question (thanks to Mark44):
Let $I_n = \int_0^{\pi/2} x \cos^n(x) dx$, with $n \ge 0$
i) Show that $n I_n = (n - 1)I_{n-2} - n^{- 1}$, for $n \ge 2$.
ii) Find the exact value of $I_3$.

I would try integration by parts twice, starting with $u = \cos^n(x), dv = xdx$. After the second integration by parts, you should have an equation that you can solve algebraically for $I_n$ in terms of $I_{n - 2}$ and other terms.