- #1
AndersCarlos
- 31
- 0
Homework Statement
a) Show that:
[tex]\int_{0}^{\pi} xf(sin (x))dx = \frac{\pi}{2}\int_{0}^{\pi} f(sin (x))dx[/tex]
[Hint: u = π - x]
b) Use part a) to deduce the formula:
[tex]\int_{0}^{\pi} \frac{xsin(x)}{1 + cos^2 (x)} dx = \pi\int_{0}^{1} \frac{dx}{1 + x^2}[/tex]
Homework Equations
[tex]\int_{a}^{b} f(x)dx = \frac{1}{k}\int_{ka}^{kb} f(\frac{x}{k})[/tex]
for any constant 'k'.
The Attempt at a Solution
a)
Tried to imagine where I could apply the hint and used the property: -sin(x) = sin(π-x) in the process of substitution, but no progress.
b)
I used the only relevant equation to convert the interval [0,1] to [0, π] in the right side, considering k = π, however I was not able to progress much after this.