$\int_{-1}^1 \sin^5 x \cos^4 x \, dx = $ . . . . . .

The value of $\int_{0}^{\pi /2} \frac{e^{x^2}}{e^{x^2} + e^{(\pi /2 - x)^2}} dx$ is

If $\int_0^{\frac{\pi}{4}} \frac{\sin^2 x}{1+\sin x \cos x} dx = \frac{1}{a} \log_e\left(\frac{a}{3}\right) + \frac{\pi}{b \sqrt{3}}$,where $a, b \in N$,then $a+b$ is equal to ....................

The value of $\int_0^{\frac{\pi}{2}} \frac{dx}{1+\tan^3 x}$ is:

Suppose the limit $L = \lim_{n \rightarrow \infty} \sqrt{n} \int_0^1 \frac{1}{(1+x^2)^n} dx$ exists and is larger than $\frac{1}{2}$. Then,

$\int_{0}^{1} \tan ^{-1}\left[\frac{2 x-1}{1+x-x^{2}}\right] d x=$