If $\int_{-1}^4 f(x) dx = 4$ and $\int_2^4 \{3 - f(x)\} dx = 7$, then find the value of $\int_{-1}^2 f(x) dx$.

$\int_{-1}^1 \frac{x^3+|x|+1}{x^2+2|x|+1} dx$ is equal to

Let $g(t) = \int_{-\pi/2}^{\pi/2} \cos \left(\frac{\pi}{4} t + f(x)\right) \, dx$,where $f(x) = \log_e \left(x + \sqrt{x^2 + 1}\right)$,$x \in R$. Then which one of the following is correct?

If $f(\alpha) = \int_{1}^{\alpha} \frac{\log_{10} t}{1+t} dt, \alpha > 0$,then $f(e^{3}) + f(e^{-3})$ is equal to.

For $n \in N$,the value of $\int_{0}^{n\pi + V} \sqrt{\frac{1 + \cos 2x}{2}} dx$ is . . . (where $\frac{\pi}{2} < V < \pi$)

The function $F(x) = \int_0^x \log \left( \frac{1 - t}{1 + t} \right) \,dt$ is