The value of $b > 3$ for which $12 \int \limits_{3}^{b} \frac{1}{(x^{2}-1)(x^{2}-4)} dx = \log _{e}(\frac{49}{40})$ is equal to

Let $f(x) = \int \frac{\sqrt{x}}{(1+x)^2} dx$ $(x \geq 0)$. Then $f(3) - f(1)$ is equal to

$\int_0^{\pi /4} {\frac{{4\sin 2\theta \,d\theta }}{{{{\sin }^4}\theta + {{\cos }^4}\theta }}} = $

Let $f$ be a polynomial function such that $f(x^{2}+1)=x^{4}+5x^{2}+2$ for all $x \in \mathbb{R}.$ Then $\int_{0}^{3} f(x) dx$ is equal to

For a real number $x$,let $[x]$ denote the largest integer less than or equal to $x$ and $\{x\}=x-[x]$. The smallest possible integer value of $n$ for which $\int_1^n [x]\{x\} dx$ exceeds $2013$ is

By the definition of the definite integral,the value of $\lim _{n \rightarrow \infty}\left(\frac{1^4}{1^5+n^5}+\frac{2^4}{2^5+n^5}+\frac{3^4}{3^5+n^5}+\ldots+\frac{n^4}{n^5+n^5}\right)$ is