Definite Integration

Let $f:\mathbb R\to\mathbb R$ be given by $f(x) = (x-1)(x-2)(x-5)$. Define

$$F(x) = \int_0^xf(t)\,dt,\quad x > 0.$$

Then which of the following options is/are correct?

The value of the integral

$$\int_0^{\pi/2}\frac{3\sqrt{\cos\theta}}{\left(\sqrt{\cos\theta} + \sqrt{\sin\theta}\right)^5}\,d\theta$$

equals ______

Let $f: [0, \infty) \to \mathbb{R}$ be a continuous function such that

$$f(x) = 1 - 2x + \int_0^x e^{x-t} f(t)\, dt$$

for all $x \in [0, \infty)$. Then, which of the following statement(s) is (are) TRUE?

For each positive integer $n$, let

$$y_n = \frac{1}{n}\left((n+1)(n+2)\cdots(n+n)\right)^{\frac{1}{n}}.$$

For $x \in \mathbb{R}$, let $[x]$ be the greatest integer less than or equal to $x$. If $\displaystyle\lim_{n \to \infty} y_n = L$, then the value of $[L]$ is _____ .

The value of the integral

$$\int_0^{1/2} \frac{1 + \sqrt{3}}{\left((x+1)^2 (1-x)^6\right)^{1/4}}\, dx$$

is _____ .

Let $f: \mathbb{R} \to (0, 1)$ be a continuous function. Then, which of the following function(s) has(have) the value zero at some point in the interval $(0, 1)$?

Let $f: \mathbb{R} \to \mathbb{R}$ be a differentiable function such that $f(0) = 0$, $f\left(\frac{\pi}{2}\right) = 3$ and $f'(0) = 1$. If

$$g(x) = \int_x^{\frac{\pi}{2}} \left[f'(t)\operatorname{cosec} t - \cot t\operatorname{cosec} t\, f(t)\right] dt$$

for $x \in \left(0, \frac{\pi}{2}\right]$, then $\lim_{x \to 0} g(x) =$

If $g(x) = \displaystyle\int_{\sin x}^{\sin(2x)} \sin^{-1}(t)\, dt$, then

If $I = \displaystyle\sum_{k=1}^{98}\int_k^{k+1}\frac{k + 1}{x(x + 1)}\, dx$, then

The integral $\displaystyle\int_{\pi/4}^{3\pi/4}\frac{dx}{1+\cos x}$ is equal to :

The integral $\displaystyle\int_{\pi/12}^{\pi/4}\frac{8\cos2x}{(\tan x+\cot x)^3}\,dx$ equals :

If $\displaystyle\int_1^2\frac{dx}{(x^2-2x+4)^{3/2}}=\frac k{k+5}$, then k is equal to :

The total number of distinct $x\in[0,1]$ for which $\displaystyle\int_0^x\frac{t^2}{1+t^4}\,dt=2x-1$ is

The value of $\displaystyle\int_{-\pi/2}^{\pi/2}\frac{x^2\cos x}{1+e^x}\,dx$ is equal to

Let $f(x)=\displaystyle\lim_{n\to\infty}\left(\frac{n^n(x+n)\left(x+\frac n2\right)\cdots\left(x+\frac nn\right)}{n!\,(x^2+n^2)\left(x^2+\frac{n^2}{4}\right)\cdots\left(x^2+\frac{n^2}{n^2}\right)}\right)^{x/n}$, for all $x>0$. Then

$\displaystyle\lim_{n\to\infty}\left(\frac{(n+1)(n+2)\dots3n}{n^{2n}}\right)^{1/n}$ is equal to :

If $2\displaystyle\int_0^1\tan^{-1}x\,dx=\int_0^1\cot^{-1}(1-x+x^2)\,dx$, then $\displaystyle\int_0^1\tan^{-1}(1-x+x^2)\,dx$ is equal to :

The value of the integral $\displaystyle\int_4^{10}\frac{[x^2]\,dx}{[x^2-28x+196]+[x^2]}$, where [x] denotes the greatest integer less than or equal to x, is :

For x ∈ R, x ≠ 0, if y(x) is a differentiable function such that $x\displaystyle\int_1^xy(t)\,dt=(x+1)\int_1^xt\,y(t)\,dt$, then y(x) equals :

(where C is a constant.)

Let $f:\mathbb R\to\mathbb R$ be a function defined by $f(x)=\begin{cases}[x],&x\le2\\0,&x>2\end{cases}$, where $[x]$ is the greatest integer less than or equal to $x$. If $I=\displaystyle\int_{-1}^{2}\frac{xf(x^2)}{2+f(x+1)}\,dx$, then the value of $(4I-1)$ is