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Belarus
Тогда у нас будет sinx = sin(t+pi/2)= cos t; cos^2x = sin^2t.
Интеграл будет равен сумме интегралов
I1 = integrate(t*cost/1+sin^2t) from -pi/2 to pi/2 = 0, так как нечетная функция;
I2 = pi/2*integrate (cost /1+sin^2t) dt from -pi/2 to pi/2
Второй интеграл легко посчитать заменой z = sint:
integrate (cost /1+sin^2t)dt = integrate(dz/1+z^2) from -1 to 1 = arctg(1)-arctg(-1) = pi/2, то есть I2 = pi^2/4
Общий ответ pi^2/4