Leibniz rule

Let $f$ be a twice differentiable function such that $f(x)=\displaystyle\int_0^x\tan(t-x)\,dt-\int_0^xf(t)\tan t\,dt$, $x\in\left(-\dfrac\pi2,\dfrac\pi2\right)$. Then $f''\left(\dfrac\pi6\right)+12f'\left(-\dfrac\pi6\right)+f\left(\dfrac\pi6\right)$ 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

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

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

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

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 $F(x)=\displaystyle\int_x^{x^2+\frac\pi6}2\cos^2t\,dt$ for all $x\in\mathbb R$ and $f:\left[0,\frac12\right]\to[0,\infty)$ be a continuous function. For $a\in\left[0,\frac12\right]$, if $F'(a)+2$ is the area of the region bounded by $x=0$, $y=0$, $y=f(x)$ and $x=a$, then $f(0)$ is

Let $f:\mathbb R\to\mathbb R$ be a continuous odd function, which vanishes exactly at one point and $f(1)=\dfrac12$. Suppose that $F(x)=\displaystyle\int_{-1}^xf(t)\,dt$ for all $x\in[-1,2]$ and $G(x)=\displaystyle\int_{-1}^xt|f(f(t))|\,dt$ for all $x\in[-1,2]$. If $\displaystyle\lim_{x\to1}\frac{F(x)}{G(x)}=\frac1{14}$, then the value of $f\left(\dfrac12\right)$ is

Let $f:(-1,1)\to\mathbb R$ be a continuous function. If $\displaystyle\int_0^{\sin x}f(t)\,dt=\frac{\sqrt3}2x$, then $f\left(\dfrac{\sqrt3}2\right)$ is equal to :

Let $f:(0,\infty)\to\mathbb R$ be given by
$$f(x)=\int_{1/x}^{x}e^{-\left(t+\frac1t\right)}\frac{dt}t.$$
Then

Let $f:[0,2]\to\mathbb R$ be a function which is continuous on $[0,2]$ and is differentiable on $(0,2)$ with $f(0)=1$. Let
$$F(x)=\int_0^{x^2}f(\sqrt t)\,dt$$
for $x\in[0,2]$. If $F'(x)=f'(x)$ for all $x\in(0,2)$, then $F(2)$ equals

Let
$$g(x)=\int_0^{e^x}\frac{f'(t)}{1+t^2}dt.$$
Which of the following is true?

$$\lim_{x\to\frac\pi4}\frac{\displaystyle\int_2^{\sec^2x}f(t)\,dt}{x^2-\frac{\pi^2}{16}}$$
equals