Definite Integration

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

If
$$\alpha=\int_0^1\left(e^{9x+3\tan^{-1}x}\right)\left(\frac{12+9x^2}{1+x^2}\right)dx$$
where $\tan^{-1}x$ takes only principal values, then the value of $\left(\log_e|1+\alpha|-\dfrac{3\pi}4\right)$ 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

The option(s) with the values of $a$ and $L$ that satisfy the following equation is(are)
$$\frac{\displaystyle\int_0^{4\pi}e^t(\sin^6at+\cos^4at)\,dt}{\displaystyle\int_0^{\pi}e^t(\sin^6at+\cos^4at)\,dt}=L?$$

Let $f(x)=7\tan^8x+7\tan^6x-3\tan^4x-3\tan^2x$ for all $x\in\left(-\dfrac\pi2,\dfrac\pi2\right)$. Then the correct expression(s) is(are)

Let $f'(x)=\dfrac{192x^3}{2+\sin^4\pi x}$ for all $x\in\mathbb R$ with $f\left(\dfrac12\right)=0$. If $m\le\displaystyle\int_{1/2}^1f(x)\,dx\le M$, then the possible values of $m$ and $M$ are

If $\displaystyle\int_1^3x^2F'(x)\,dx=-12$ and $\displaystyle\int_1^3x^3F''(x)\,dx=40$, then the correct expression(s) is(are)

The integral
$$\int_2^4\frac{\log x^2}{\log x^2+\log(36-12x+x^2)}dx$$
is equal to :

For $x>0$, let $f(x)=\displaystyle\int_1^x\frac{\log t}{1+t}dt$. Then $f(x)+f\left(\dfrac1x\right)$ is equal to :

Let $f:\mathbb R\to\mathbb R$ be a function such that $f(2-x)=f(2+x)$ and $f(4-x)=f(4+x)$, for all $x\in\mathbb R$ and $\displaystyle\int_0^2f(x)\,dx=5$. Then the value of $\displaystyle\int_{10}^{50}f(x)\,dx$ 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 $f:[a,b]\to[1,\infty)$ be a continuous function and let $g:\mathbb R\to\mathbb R$ be defined as
$$g(x)=\begin{cases}0&\text{if }x<a,\\\displaystyle\int_a^xf(t)\,dt&\text{if }a\le x\le b,\\\displaystyle\int_a^bf(t)\,dt&\text{if }x>b.\end{cases}$$
Then

The value of
$$\int_0^14x^3\left\{\frac{d^2}{dx^2}(1-x^2)^5\right\}dx$$
is

The function $y=f(x)$ is the solution of the differential equation
$$\frac{dy}{dx}+\frac{xy}{x^2-1}=\frac{x^4+2x}{\sqrt{1-x^2}}$$
in $(-1,1)$ satisfying $f(0)=0$. Then
$$\int_{-\frac{\sqrt3}{2}}^{\frac{\sqrt3}{2}}f(x)\,dx$$
is

The following integral
$$\int_{\pi/4}^{\pi/2}(2\operatorname{cosec}x)^{17}\,dx$$
is equal to

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

The value of $g\left(\frac12\right)$ is

The value of $g'\left(\frac12\right)$ is

Match List I with List II and select the correct answer using the code given below the lists.