- #1

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## Homework Statement

a) Solve:[tex]^{Pi}_{0}[/tex][tex]\int[/tex][tex]\frac{sin(x)}{1 + cos²x}[/tex]dx

b) Proof that for each f, continuous in [0, a], [tex]^{a}_{0}[/tex][tex]\int[/tex][tex]{f(x)}[/tex]dx = [tex]^{a}_{0}[/tex][tex]\int[/tex][tex]{f(a-x)}[/tex]dx

c) Use a and b to solve [tex]^{Pi}_{0}[/tex][tex]\int[/tex][tex]\frac{x sin(x)}{1 + cos²x}[/tex]dx

## Homework Equations

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## The Attempt at a Solution

a) t = cos(x)

dt/dx = sin(x)

dt = sin(x)*dx

[tex]^{Pi}_{0}[/tex][tex]\int[/tex][tex]\frac{sin(x)}{1 + cos²x}[/tex]dx

= [tex]^{1}_{-1}[/tex][tex]\int[/tex][tex]\frac{dt}{1 + t²}[/tex]dt

= arctan(1)-arctan(-1) = Pi/2

b) t = a - x

dt/dx = -1

-dt = dx

[tex]^{0}_{a}[/tex][tex]\int[/tex][tex]{-f(t)}[/tex]dt

= [tex]^{a}_{0}[/tex][tex]\int[/tex][tex]{f(t)}[/tex]dt

= [tex]^{a}_{0}[/tex][tex]\int[/tex][tex]{f(x)}[/tex]dx

c) I have no idea to start this should I replace x with Pi-x, I tried this but i'm not getting any further