Definite Integration

Let $f(x)=\begin{cases}\dfrac13, & x\le\dfrac\pi2\\[2mm] \dfrac{b(1-\sin x)}{(\pi-2x)^2}, & x>\dfrac\pi2\end{cases}$. If $f$ is continuous at $x=\pi/2$, then the value of $\displaystyle\int_0^{3b-6}\left|x^2+2x-3\right|dx$ is :

If $\displaystyle\int_{\pi/6}^{\pi/4}\left(\cot\left(x-\frac\pi3\right)\cot\left(x+\frac\pi3\right)+1\right)dx=\alpha\log_e\left(\sqrt3-1\right)$, then $9\alpha^2$ is equal to ________.

Let $\alpha$ and $\beta$ be the real numbers such that

$$\lim_{x\to0}\frac{1}{x^3}\left(\frac{\alpha}{2}\int_0^x\frac{1}{1-t^2}\,dt + \beta x\cos x\right) = 2.$$

Then the value of $\alpha + \beta$ is _____.

If

$$\alpha = \int_{1/2}^{2}\frac{\tan^{-1}x}{2x^2 - 3x + 2}\,dx,$$

then the value of $\sqrt7\tan\left(\dfrac{2\alpha\sqrt7}{\pi}\right)$ is _____.

(Here, the inverse trigonometric function $\tan^{-1}x$ assumes values in $\left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)$.)

Let the function $f : [1, \infty) \to \mathbb{R}$ be defined by

$$f(t) = \begin{cases} (-1)^{n+1}\,2, & \text{if } t = 2n-1,\ n \in \mathbb{N}, \\ \dfrac{(2n+1-t)}{2}f(2n-1) + \dfrac{(t-(2n-1))}{2}f(2n+1), & \text{if } 2n-1 < t < 2n+1,\ n \in \mathbb{N}. \end{cases}$$

Define $g(x) = \displaystyle\int_1^x f(t)\,dt$, $x \in (1, \infty)$. Let $\alpha$ denote the number of solutions of the equation $g(x) = 0$ in the interval $(1, 8]$ and $\beta = \displaystyle\lim_{x\to 1+}\frac{g(x)}{x-1}$. Then the value of $\alpha + \beta$ is equal to _____.

The value of $\displaystyle 2\int_0^{\pi/2} f(x)g(x)\,dx - \int_0^{\pi/2} g(x)\,dx$ is _____.

The value of $\displaystyle\frac{16}{\pi^3}\int_0^{\pi/2} f(x)g(x)\,dx$ is _____.

Let $f : [0,1] \to [0,1]$ be the function defined by $f(x) = \dfrac{x^3}{3} - x^2 + \dfrac59x + \dfrac{17}{36}$. Consider the square region $S = [0,1]\times[0,1]$. Let $G = \{(x,y) \in S : y > f(x)\}$ be called the green region and $R = \{(x,y) \in S : y < f(x)\}$ be called the red region. Let $L_h = \{(x,h) \in S : x \in [0,1]\}$ be the horizontal line drawn at a height $h \in [0,1]$. Then which of the following statements is(are) true?

Let $f : (0,1) \to \mathbb{R}$ be the function defined as $f(x) = \sqrt n$ if $x \in \left[\dfrac{1}{n+1}, \dfrac1n\right)$ where $n \in \mathbb{N}$. Let $g : (0,1) \to \mathbb{R}$ be a function such that $\displaystyle\int_{x^2}^{x}\sqrt{\frac{1-t}{t}}\,dt < g(x) < 2\sqrt x$ for all $x \in (0,1)$. Then $\displaystyle\lim_{x\to0}f(x)g(x)$

Let $f : [1,\infty) \to \mathbb{R}$ be a differentiable function such that $f(1) = \dfrac13$ and $3\displaystyle\int_1^x f(t)\,dt = x\,f(x) - \dfrac{x^3}{3}$, $x \in [1,\infty)$. Let $e$ denote the base of the natural logarithm. Then the value of $f(e)$ is

For $x \in \mathbb{R}$, let $\tan^{-1}(x) \in \left(-\dfrac\pi2, \dfrac\pi2\right)$. Then the minimum value of the function $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = \displaystyle\int_0^{x\tan^{-1}x} \frac{e^{(t-\cos t)}}{1+t^{2023}}\,dt$ is _____.

Consider the equation
$$\int_1^e \frac{(\log_e x)^{1/2}}{x\left(a - (\log_e x)^{3/2}\right)^2}\,dx = 1, \qquad a \in (-\infty, 0)\cup(1,\infty).$$
Which of the following statements is/are TRUE?

The greatest integer less than or equal to
$$\int_1^2 \log_2(x^3+1)\,dx + \int_1^{\log_2 9} (2^x - 1)^{\frac13}\,dx$$
is _____.

For positive integer $n$, define
$$f(n) = n + \frac{16 + 5n - 3n^2}{4n + 3n^2} + \frac{32 + n - 3n^2}{8n + 3n^2} + \frac{48 - 3n - 3n^2}{12n + 3n^2} + \cdots + \frac{25n - 7n^2}{7n^2}.$$
Then, the value of $\displaystyle\lim_{n\to\infty} f(n)$ is equal to

Let $f(\theta)=\sin\theta+\displaystyle\int_{-\pi/2}^{\pi/2}(\sin\theta+t\cos\theta)f(t)\,dt$. Then the value of $\left|\displaystyle\int_0^{\pi/2}f(\theta)\,d\theta\right|$ is ______.

Let $\displaystyle\max_{0\le x\le2}\left\{\frac{9-x^2}{5-x}\right\}=\alpha$ and $\displaystyle\min_{0\le x\le2}\left\{\frac{9-x^2}{5-x}\right\}=\beta$.

If $\displaystyle\int_{\beta-\frac83}^{2\alpha-1}\max\left\{\frac{9-x^2}{5-x},x\right\}dx=\alpha_1+\alpha_2\log_e\left(\frac8{15}\right)$ then $\alpha_1+\alpha_2$ is equal to ______.

The value of the integral
$$\int_{-\pi/2}^{\pi/2}\frac{dx}{\left(1+e^x\right)\left(\sin^6x+\cos^6x\right)}$$
is equal to

$$\lim_{n\to\infty}\left(\frac{n^2}{\left(n^2+1\right)(n+1)}+\frac{n^2}{\left(n^2+4\right)(n+2)}+\frac{n^2}{\left(n^2+9\right)(n+3)}+\dots+\frac{n^2}{\left(n^2+n^2\right)(n+n)}\right)$$
is equal to

The value of $\displaystyle\int_0^\pi\frac{e^{\cos x}\sin x}{\left(1+\cos^2x\right)\left(e^{\cos x}+e^{-\cos x}\right)}dx$ is equal to :

If $b_n=\displaystyle\int_0^{\frac\pi2}\frac{\cos^2nx}{\sin x}dx$, $n\in\mathbb N$, then