Definite Integration

The integral $\displaystyle\int_0^{\pi/2}\frac1{3+2\sin x+\cos x}\,dx$ is equal to :

If $f(\alpha)=\displaystyle\int_1^\alpha\frac{\log_{10}t}{1+t}\,dt$, α > 0, then $f(e^3)+f(e^{-3})$ is equal to :

If [t] denotes the greatest integer ≤ t, then the value of $\displaystyle\int_0^1\left[2x-\left|3x^2-5x+2\right|+1\right]dx$ is :

If $y(x)=\tan x-\displaystyle\int_0^{2x}(x^2-xt+1)\sin t\,dt$, then at $x=\dfrac\pi3$, the value of $2y'''-y''$ is equal to

Let $A_k=\displaystyle\int_{k\pi}^{(k+1)\pi}e^{-x}\sin x\,dx$, k = 0, 1, 2, ..., 20. Then $\displaystyle\sum_{k=0}^{20}|A_k|$ is equal to

$\displaystyle\lim_{n\to\infty}\sum_{k=1}^n\frac{n(n^2+k^2)}{(n^4+n^2k^2+k^4)}$ is equal to

Let $f: \left[-\frac\pi2, \frac\pi2\right] \to \mathbb R$ be a continuous function such that
$$f(0) = 1 \quad\text{and}\quad \int_0^{\pi/3} f(t)\,dt = 0$$
Then which of the following statements is (are) TRUE?

The value of $2m_1 + 3n_1 + m_1n_1$ is ____.

The value of $\dfrac{16S_1}{\pi}$ is ____.

The value of $\dfrac{48S_2}{\pi^2}$ is ____.

Which of the following statements is TRUE?

Which of the following statements is TRUE?

For any real number $x$, let $[x]$ denote the largest integer less than or equal to $x$. If
$$I = \int_0^{10}\left[\sqrt{\frac{10x}{x + 1}}\right]dx,$$
then the value of $9I$ is ____.

$\displaystyle\lim_{x\to0}\frac{\int_0^{x^2}\left(\sin\sqrt t\right)dt}{x^3}$ is equal to :

If $\displaystyle\int_{-a}^a(|x|+|x-2|)dx=22$, (a > 2) and [x] denotes the greatest integer ≤ x, then $\displaystyle\int_a^{-a}(x+[x])dx$ is equal to _______.

Let the functions $f:\mathbb R\to\mathbb R$ and $g:\mathbb R\to\mathbb R$ be defined by

$$f(x) = e^{x-1} - e^{-|x-1|}\quad\text{and}\quad g(x) = \frac12\left(e^{x-1} + e^{1-x}\right).$$

Then the area of the region in the first quadrant bounded by the curves $y = f(x)$, $y = g(x)$ and $x = 0$ is

Which of the following inequalities is/are TRUE?

Let $f:\mathbb R\to\mathbb R$ be a differentiable function such that its derivative $f'$ is continuous and $f(\pi) = -6$. If $F:[0,\pi]\to\mathbb R$ is defined by $F(x) = \int_0^x f(t)\,dt$, and if

$$\int_0^\pi\left(f'(x) + F(x)\right)\cos x\,dx = 2,$$

then the value of $f(0)$ is ______

If

$$I = \frac2\pi\int_{-\pi/4}^{\pi/4}\frac{dx}{(1 + e^{\sin x})(2 - \cos2x)}$$

then $27I^2$ equals ______

For $a\in\mathbb R$, $|a| > 1$, let

$$\lim_{n\to\infty}\left(\frac{1 + \sqrt[3]2 + \cdots + \sqrt[3]n}{n^{7/3}\left(\frac1{(an+1)^2} + \frac1{(an+2)^2} + \cdots + \frac1{(an+n)^2}\right)}\right) = 54.$$

Then the possible value(s) of $a$ is/are