The positive value of $x$ satisfying the equation $\int_x^1(1-t) dt = \frac{1}{2}$ is

The value of the integral $\int \limits_1^2 \left(\frac{t^4+1}{t^6+1}\right) dt$ is $..........$.

On the interval $\left[ \frac{5\pi}{3}, \frac{7\pi}{4} \right]$,the greatest value of the function $f(x) = \int_{5\pi/3}^x (6\cos t - 2\sin t) \, dt$ is:

Let $f(x) = 2 + |x| - |x - 1| + |x + 1|$,$x \in R$. Consider:
$(S1): f^{\prime}\left(-\frac{3}{2}\right) + f^{\prime}\left(-\frac{1}{2}\right) + f^{\prime}\left(\frac{1}{2}\right) + f^{\prime}\left(\frac{3}{2}\right) = 4$
$(S2): \int_{-2}^{2} f(x) dx = 12$
Then,

Which of the following is true?

Evaluate the definite integral $\int_{0}^{\frac{\pi}{4}} \tan x \,dx$.