Definite Integration

The value of $b>3$ for which $12\displaystyle\int_3^b\frac1{\left(x^2-1\right)\left(x^2-4\right)}dx=\log_e\left(\frac{49}{40}\right)$, is equal to ______.

Let $f(x)=\max\{|x+1|,|x+2|,\dots,|x+5|\}$. Then $\displaystyle\int_{-6}^0f(x)\,dx$ is equal to ______.

The value of the integral $\dfrac{48}{\pi^4}\displaystyle\int_0^\pi\left(\frac{3\pi x^2}2-x^3\right)\frac{\sin x}{1+\cos^2x}dx$ is equal to ______.

The integral $\dfrac{24}\pi\displaystyle\int_0^{\sqrt2}\frac{\left(2-x^2\right)dx}{\left(2+x^2\right)\sqrt{4+x^4}}$ is equal to ______.

The value of the integral $\displaystyle\int_{-2}^2\frac{\left|x^3+x\right|}{\left(e^{x|x|}+1\right)}dx$ is equal to :

If $m$ and $n$ respectively are the number of local maximum and local minimum points of the function $f(x)=\displaystyle\int_0^{x^2}\frac{t^2-5t+4}{2+e^t}dt$, then the ordered pair $(m,n)$ is equal to

Let $f$ be a differentiable function in $\left(0,\dfrac\pi2\right)$. If $\displaystyle\int_{\cos x}^1t^2f(t)\,dt=\sin^3x+\cos x$, then $\dfrac1{\sqrt3}f'\left(\dfrac1{\sqrt3}\right)$ is equal to

The integral $\displaystyle\int_0^1\frac1{7^{\left[\frac1x\right]}}dx$, where $[\cdot]$ denotes the greatest integer function, is equal to

Let $y=y(x)$ be the solution of the differential equation $\left(1-x^2\right)dy=\left(xy+\left(x^3+2\right)\sqrt{1-x^2}\right)dx$, $-1<x<1$, and $y(0)=0$. If $\displaystyle\int_{-1/2}^{1/2}\sqrt{1-x^2}\,y(x)\,dx=k$, then $k^{-1}$ is equal to ______.

Let $[t]$ denote the greatest integer less than or equal to $t$. Then, the value of the integral $\displaystyle\int_0^1\left[-8x^2+6x-1\right]dx$ is equal to

Let $f:\mathbf R\to\mathbf R$ be a differentiable function such that $f\left(\dfrac\pi4\right)=\sqrt2$, $f\left(\dfrac\pi2\right)=0$ and $f'\left(\dfrac\pi2\right)=1$ and let $g(x)=\displaystyle\int_x^{\pi/4}\left(f'(t)\sec t+\tan t\sec t\,f(t)\right)dt$ for $x\in\left[\dfrac\pi4,\dfrac\pi2\right)$. Then $\displaystyle\lim_{x\to\left(\frac\pi2\right)^-}g(x)$ is equal to :

Let $f:\mathbf R\to\mathbf R$ be a continuous function satisfying $f(x)+f(x+k)=n$, for all $x\in\mathbf R$ where $k>0$ and $n$ is a positive integer. If $I_1=\displaystyle\int_0^{4nk}f(x)dx$ and $I_2=\displaystyle\int_{-k}^{3k}f(x)dx$, then :

Let the slope of the tangent to a curve $y=f(x)$ at $(x,y)$ be given by $2\tan x(\cos x-y)$. If the curve passes through the point $\left(\frac\pi4,0\right)$, then the value of $\displaystyle\int_0^{\pi/2}y\,dx$ is equal to :

Let $f:\mathbf R\to\mathbf R$ be a function defined by :
$$f(x)=\begin{cases}\displaystyle\max_{t\le x}\left\{t^3-3t\right\}, & x\le2\\ x^2+2x-6, & 2<x<3\\ [x-3]+9, & 3\le x\le5\\ 2x+1, & x>5\end{cases}$$
where $[t]$ is the greatest integer less than or equal to $t$. Let $m$ be the number of points where $f$ is not differentiable and $I=\displaystyle\int_{-2}^2f(x)\,dx$. Then the ordered pair $(m,I)$ is equal to :

$\displaystyle\int_0^5\cos\left(\pi\left(x-\left[\frac x2\right]\right)\right)dx$,

where $[t]$ denotes greatest integer less than or equal to $t$, is equal to :

Let $f$ be a real valued continuous function on $[0,1]$ and $f(x)=x+\displaystyle\int_0^1(x-t)f(t)\,dt$. Then, which of the following points $(x,y)$ lies on the curve $y=f(x)$?

$\displaystyle\lim_{n\to\infty}\sum_{r=1}^n\frac r{2r^2-7rn+6n^2}$ is equal to :

Let $f(t)=\displaystyle\int_0^te^{x^3}\left(\frac{x^8}{\left(x^6+2x^3+2\right)^2}\right)dx$. If $f(1)+f'(1)=\alpha e-\dfrac16$, then the value of $150\alpha$ is equal to ______.

For any real number $x$, let $[x]$ denote the largest integer less than equal to $x$. Let $f$ be a real valued function defined on the interval $[-10,10]$ by $f(x)=\begin{cases}x-[x], & \text{if }[x]\text{ is odd}\\1+[x]-x, & \text{if }[x]\text{ is even}\end{cases}$. Then the value of $\dfrac{\pi^2}{10}\displaystyle\int_{-10}^{10}f(x)\cos\pi x\,dx$ is :

If $\displaystyle\lim_{n\to\infty}\frac{(n+1)^{k-1}}{n^{k+1}}\left[(nk+1)+(nk+2)+\dots+(nk+n)\right]=33\cdot\lim_{n\to\infty}\frac1{n^{k+1}}\cdot\left[1^k+2^k+3^k+\dots+n^k\right]$, then the integral value of k is equal to ______.