Definite Integration

For any real number $x$, let $\left[x\right]$ denote the largest integer less than or equal to $x$. Let $f$be a real-valued function defined on the interval $\left[-10,10\right]$ by $f\left(x\right)=\left{\begin{matrix}…

Let$S=\left(-1,\infty\right)$ and$f:S\rightarrow\,\mathbb{R}$ be defined as…

If $(a,b)$ be the orthocentre of the triangle whose vertices are $(1,2),(2,3)$ and $(3,1)$, and…

The integral$\underset{0}{\overset{\pi\,}{\int\,}}\sqrt{1+4\text{sin}^{2}\frac{x}{2}-4\text{sin}\frac{x}{2}}dx$ equals

Let $I=\int\,_{a}^{b}\left(x^{4}-2x^{2}\right)dx.$If $I$ is minimum then the ordered pair $(a,b)$ is

Let $f$ be a twice differentiable function such that $f(x)=\int_{0}^{x}\tan(t-x)dt-\int_{0}^{x}f(t)\tan t\,dt$, $x \in \left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$. Then…

Let a differentiable function $f$ satisfy the equation $\int_{0}^{36} f\left(\frac{t x}{36}\right) d t=4 \alpha f(x)$. If $y=f(x)$ is a standard parabola passing through the points $(2,1)$ and $(-4, \beta)$, then…

If $\int\,_{-a}^{a}\left(\left|x\right|+\left|x-2\right|\right)dx=22,\left(a>2\right)$and $\left[x\right]$ denotes the greatest integer$\leq\,x$, then $\int\,_{a}^{-a}\left(x+\left[x\right]\right)dx$ is equal to

If $\int\,_{0}^{1}\left(x^{21}+x^{14}+x^{7}\right)\left(2x^{14}+3x^{7}+6\right)^{1/7}dx=\frac{1}{l}\left(11\right)^{m/n}$where $l,m,n\in\,N,m$ and $n$ are co-prime then $l+m+n$ is equal to _____ .

Let $f\left(x\right)$be a function satisfying $f\left(x\right)+f\left(\pi\,-x\right)=\pi\,^{2},\forall\,x\in\,\mathbb{R}$. Then $\int\,_{0}^{\pi\,}f\left(x\right)\sin\,xdx$is equal to

$I=\int\,_{\frac{\pi\,}{4}}^{\frac{\pi\,}{3}}\left(\frac{8\sin\,x-\sin\,2x}{x}\right)dx$. Then

For $m,n>0$, let $\alpha\,\left(m,n\right)=\int\,_{0}^{2}t^{m}\left(1+3t\right)^{n}dt$. If ,$11\alpha\,\left(10,6\right)+18\alpha\,\left(11,5\right)=p\left(14\right)^{6}$, then $p$ is equal to