Definite Integration

The integral $\displaystyle\int_0^\pi\sqrt{1+4\sin^2\frac x2-4\sin\frac x2}\,dx$ equals :

The integral $\displaystyle\int_0^{1/2}\frac{\ln(1+2x)}{1+4x^2}\,dx$, equals :

If for $n\ge1$, $P_n=\displaystyle\int_1^e(\log x)^n\,dx$, then $P_{10}-90P_8$ is equal to :

If [ ] denotes the greatest integer function, then the integral $\displaystyle\int_0^\pi[\cos x]\,dx$ is equal to :

If for a continuous function $f(x)$, $\displaystyle\int_{-\pi}^t(f(x)+x)\,dx=\pi^2-t^2$, for all $t\ge-\pi$, then $f\left(-\dfrac\pi3\right)$ is equal to :

Let function F be defined as $F(x)=\displaystyle\int_1^x\frac{e^t}t\,dt$, $x>0$ then the value of the integral $\displaystyle\int_1^x\frac{e^t}{t+a}\,dt$, where $a>0$, is :

Let $f:\left[\frac12,1\right]\to\mathbb R$ (the set of all real numbers) be a positive, non-constant and differentiable function such that $f'(x)<2f(x)$ and $f\left(\frac12\right)=1$. Then the value of $\displaystyle\int_{1/2}^1f(x)\,dx$ lies in the interval

For $a\in\mathbb R$ (the set of all real numbers), $a\ne-1$,
$$\lim_{n\to\infty}\frac{1^a+2^a+\cdots+n^a}{(n+1)^{a-1}\left[(na+1)+(na+2)+\cdots+(na+n)\right]}=\frac1{60}.$$
Then $a=$

Let $S$ be the area of the region enclosed by $y=e^{-x^2}$, $y=0$, $x=0$, and $x=1$. Then

The value of the integral $\displaystyle\int_{-\pi/2}^{\pi/2}\left(x^2+\ln\frac{\pi+x}{\pi-x}\right)\cos x\,dx$ is

Which of the following is true?

If $f(x)=\displaystyle\int_0^x e^{t^2}(t-2)(t-3)\,dt$ for all $x\in(0,\infty)$, then

The value of
$$\int_{\sqrt{\ln2}}^{\sqrt{\ln3}}\frac{x\sin x^2}{\sin x^2+\sin\left(\ln6-x^2\right)}\,dx$$
is

Let $f:[1,\infty)\to[2,\infty)$ be a differentiable function such that $f(1)=2$. If
$$6\int_1^xf(t)\,dt=3xf(x)-x^3$$
for all $x\ge1$, then the value of $f(2)$ is

Let $f:[-1,2]\to[0,\infty)$ be a continuous function such that $f(x)=f(1-x)$ for all $x\in[-1,2]$. Let $R_1=\displaystyle\int_{-1}^2xf(x)\,dx$, and $R_2$ be the area of the region bounded by $y=f(x)$, $x=-1$, $x=2$, and the $x$-axis. Then

Match the statements given in Column I with the values given in Column II

The value of $\displaystyle\lim_{x\to0}\frac1{x^3}\int_0^x\frac{t\ln(1+t)}{t^4+4}\,dt$ is

Let f be a real-valued function defined on the interval $(0,\infty)$ by $f(x)=\ln x+\displaystyle\int_0^x\sqrt{1+\sin t}\,dt$. Then which of the following statement(s) is (are) true?

The value(s) of $\displaystyle\int_0^1\frac{x^4(1-x)^4}{1+x^2}\,dx$ is (are)

For any real number x, let [x] denote the largest integer less than or equal to x. Let f be a real valued function defined on the interval [−10, 10] by
$$f(x)=\begin{cases}x-[x]&\text{if }[x]\text{ is odd,}\\1+[x]-x&\text{if }[x]\text{ is even.}\end{cases}$$
Then the value of $\dfrac{\pi^2}{10}\displaystyle\int_{-10}^{10}f(x)\cos\pi x\,dx$ is