pi-pi-xsin-x-1-x-2-dx-

Question Number 142656 by qaz last updated on 03/Jun/21

\int_{- π}^{π} \frac{xsin x}{1 + x^{2}} dx = ?

Answered by Ar Brandon last updated on 03/Jun/21

ξ (a) = \int_{0}^{π} \frac{xsin (ax)}{1 + x^{2}} dx \Rightarrow ξ^{'} (a) = \int_{0}^{π} \frac{x^{2} \cos (ax)}{1 + x^{2}} dx

ξ^{'} (a) = \int_{0}^{π} \cos (ax) dx - \int_{0}^{π} \frac{\cos (ax)}{1 + x^{2}} dx = - \int_{0}^{π} \frac{\cos (ax)}{1 + x^{2}} dx

ξ^{″} (a) = \int_{0}^{π} \frac{xsin (ax)}{1 + x^{2}} dx = ξ (a) \Rightarrow ξ^{″} (a) - ξ (a) = 0

ξ (a) = α e^{a} + β e^{- a}, ξ (0) = α + β = 0 \dots eqn (1)

ξ^{'} (a) = α e^{a} - β e^{- a}, ξ^{'} (0) = α - β = π - \tan^{- 1} (π) \dots eqn (2)

eqn (1) & eqn (2) \Rightarrow 2 α = π - \tan^{- 1} (π), 2 β = \tan^{- 1} (π) - π

\Rightarrow ξ (a) = \frac{1}{2} (π - \tan^{- 1} (π)) (e^{a} - e^{- a}) = (π - \tan^{- 1} (π)) \sinh (a)

\int_{- π}^{π} \frac{xsinx}{1 + x^{2}} dx = 2 ξ (1) = 2 (π - \tan^{- 1} (π)) \sinh (1)

Commented by Dwaipayan Shikari last updated on 03/Jun/21

N i c e!

Commented by Ar Brandon last updated on 03/Jun/21

Yes bro . But final answer seems not to

be correct . Still checking \dots

Commented by qaz last updated on 03/Jun/21

\int_{- π}^{π} \cos (ax) dx \neq 0

Commented by Ar Brandon last updated on 03/Jun/21

Hmmm \dots

Right answer is approximately

1, 682 984 232 . Help if possible .

Commented by mindispower last updated on 03/Jun/21

n i c e s i r

Answered by mindispower last updated on 04/Jun/21

\int_{0}^{π} \frac{2 x s i n (x)}{(x + i) (x - i)} d x

= R e {\int_{0}^{π} \frac{s i n (x)}{x + i} d x}

= R e \int_{i}^{π + i} \frac{s i n (t - i)}{t} d t

= R e \int_{i}^{π + i} \frac{s i n (t) c o s (i) - c o s (t) s i n (i)}{t}

= R e {c o s (i) \int_{i}^{π + i} \frac{s i n (t)}{t} d t - s i n (i) \int_{i}^{π + i} \frac{c o s (t)}{t} d x}

= R e {c o s (i) (S i (π + i) - S i (i)) - s i n (i) (C i (i) - C i (i + π)}

c o s (i) = c h (1), s i n (i) = i s h (1)

S i = \int_{0}^{x} \frac{s i n (t)}{t}, C i = \int_{x}^{\infty} \frac{c o s (t)}{t} d t

= \frac{1}{2} c h (1) (s i (π + i) + s i (π - i) - s i (i) - s i (- i))

- s h (1) (i C i (i) - i C i (i + π)) - s h (- i C i (- i) + i C i (π - i))