Definite Integration

$\displaystyle\lim_{n\to\infty}\frac1{2^n}\left(\frac1{\sqrt{1-\frac1{2^n}}}+\frac1{\sqrt{1-\frac2{2^n}}}+\frac1{\sqrt{1-\frac3{2^n}}}+\dots+\frac1{\sqrt{1-\frac{2^n-1}{2^n}}}\right)$ is equal to

Let $[t]$ denote the greatest integer less than or equal to $t$. Then the value of the integral $\displaystyle\int_{-3}^{101}\left([\sin(\pi x)]+e^{[\cos(2\pi x)]}\right)dx$ is equal to

Let $f$ be a twice differentiable function on $\mathbb R$. If $f'(0)=4$ and $f(x)+\displaystyle\int_0^x(x-t)f'(t)\,dt=\left(e^{2x}+e^{-2x}\right)\cos2x+\frac2ax$, then $(2a+1)^5a^2$ is equal to ______.

Let $a_n=\displaystyle\int_{-1}^n\left(1+\frac x2+\frac{x^2}3+\dots+\frac{x^{n-1}}n\right)dx$ for every $n\in\mathbb N$. Then the sum of all the elements of the set $\{n\in\mathbb N:a_n\in(2,30)\}$ is ______.

If $a=\displaystyle\lim_{n\to\infty}\sum_{k=1}^n\frac{2n}{n^2+k^2}$ and $f(x)=\sqrt{\dfrac{1-\cos x}{1+\cos x}}$, $x\in(0,1)$, then :

If $n(2n+1)\displaystyle\int_0^1\left(1-x^n\right)^{2n}dx=1177\int_0^1\left(1-x^n\right)^{2n+1}dx$, then $n\in\mathbf N$ is equal to ______.

Let a curve $y=y(x)$ pass through the point $(3,3)$ and the area of the region under this curve, above the $x$-axis and between the abscissae 3 and $x(>3)$ be $\left(\dfrac yx\right)^3$. If this curve also passes through the point $\left(\alpha,6\sqrt{10}\right)$ in the first quadrant, then $\alpha$ is equal to ______.

$\displaystyle\int_0^{20\pi}\left(|\sin x|+|\cos x|\right)^2dx$ is equal to

Let the solution curve $y=f(x)$ of the differential equation $\dfrac{dy}{dx}+\dfrac{xy}{x^2-1}=\dfrac{x^4+2x}{\sqrt{1-x^2}}$, $x\in(-1,1)$ pass through the origin. Then $\displaystyle\int_{-\frac{\sqrt3}2}^{\frac{\sqrt3}2}f(x)\,dx$ is equal to

Let $f:\mathbf R\to\mathbf R$ be a function defined as

$$f(x)=a\sin\left(\frac{\pi[x]}2\right)+[2-x],\ a\in\mathbf R,$$

where $[t]$ is the greatest integer less than or equal to $t$. If $\displaystyle\lim_{x\to-1}f(x)$ exists, then the value of $\displaystyle\int_0^4f(x)\,dx$ is equal to

Let $I=\displaystyle\int_{\pi/4}^{\pi/3}\left(\frac{8\sin x-\sin2x}{x}\right)dx$. Then

Let a function $f:\mathbf R\to\mathbf R$ be defined as :

$$f(x)=\begin{cases}\displaystyle\int_0^x(5-|t-3|)\,dt, & x>4\\ x^2+bx, & x\le4\end{cases}$$

where $b\in\mathbf R$. If $f$ is continuous at $x=4$, then which of the following statements is NOT true?

Let $f(x)=2+|x|-|x-1|+|x+1|$, $x\in\mathbf R$.

Consider

(S1) : $f'\left(-\dfrac32\right)+f'\left(-\dfrac12\right)+f'\left(\dfrac12\right)+f'\left(\dfrac32\right)=2$

(S2) : $\displaystyle\int_{-2}^2f(x)\,dx=12$

Then,

$\displaystyle\int_0^2\left(\left|2x^2-3x\right|+\left[x-\frac12\right]\right)dx$, where $[t]$ is the greatest integer function, is equal to :

Let $f(x)=\min\{[x-1],[x-2],\ldots,[x-10]\}$ where $[t]$ denotes the greatest integer $\le t$. Then $\displaystyle\int_0^{10}f(x)\,dx+\int_0^{10}(f(x))^2dx+\int_0^{10}|f(x)|\,dx$ is equal to ______.

Let $f$ be a differentiable function satisfying $f(x)=\dfrac2{\sqrt3}\displaystyle\int_0^{\sqrt3}f\left(\frac{\lambda^2x}3\right)d\lambda$, $x>0$ and $f(1)=\sqrt3$. If $y=f(x)$ passes through the point $(\alpha,6)$, then $\alpha$ is equal to ______.

The minimum value of the twice differentiable function $f(x)=\displaystyle\int_0^xe^{x-t}f'(t)\,dt-\left(x^2-x+1\right)e^x$, $x\in\mathbf R$, is :

If $\displaystyle\int_0^{\sqrt3}\frac{15x^3}{\sqrt{1+x^2+\sqrt{\left(1+x^2\right)^3}}}dx=\alpha\sqrt2+\beta\sqrt3$, where $\alpha$, $\beta$ are integers, then $\alpha+\beta$ is equal to ______.

Let $I_n(x)=\displaystyle\int_0^x\frac1{\left(t^2+5\right)^n}dt$, $n=1,2,3,\ldots$. Then :

The value of the integral $\displaystyle\int_0^{\pi/2}60\frac{\sin(6x)}{\sin x}dx$ is equal to ______.