The value of $\int_0^{2 \pi} \sqrt{1+\sin \left(\frac{x}{2}\right)} d x$ is

Let $f(x) + 2f\left(\frac{1}{x}\right) = x^2 + 5$ and $2g(x) - 3g\left(\frac{1}{x}\right) = x$ for $x > 0$. If $\alpha = \int_1^2 f(x) dx$ and $\beta = \int_1^2 g(x) dx$,then the value of $9\alpha + \beta$ is:

Let $\phi (x) = \int_{0}^{1} e^{x} e^{t} \phi (t) dt + x$. If $\phi (\ln (e^{2} - 3))$ is equal to $A$,then find the value of $A$.

The value of the definite integral $\int\limits_2^3 {\left[ {\sqrt {2x - \sqrt {5(4x - 5)} } + \sqrt {2x + \sqrt {5(4x - 5)} } } \right]} \,dx$ is:

Evaluate the definite integral: $\int_{0}^{\frac{\pi}{4}}\left(2 \sec ^{2} x+x^{3}+2\right) d x$

If $[2,6]$ is divided into four intervals of equal length,then the approximate value of $\int_2^6 \frac{1}{x^2-x} dx$ using Simpson's rule is