Definite Integration

$\(\int$ 4 $\cos$ $\left(x+\frac{\pi}{6}\right)$ $\cos$ 2 x $\cdot$ $\cos$ $\left(\frac{5$ $\pi}{6}+x\right)$ d x\)

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

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

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 $f$ 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 $[t]$ denote the largest integer less than or equal to $t$. If $\int_0^3\left(\left[x^2\right]+\left[\frac{x^2}{2}\right]\right) \mathrm{d} x=\mathrm{a}+\mathrm{b} \sqrt{2}-\sqrt{3}-\sqrt{5}+\mathrm{c}…

If$f:\mathbb{R}\rightarrow\,\mathbb{R}$be a continuous function satisfying$\int\,_{0}^{\frac{\pi\,}{2}}f\left(\sin\,2x\right)\sin\,xdx+\alpha\,\int\,_{0}^{\frac{\pi\,}{4}}f\left(\cos\,2x\right)\cos\,xdx=0$, then the…

For $x\in\,\left(-\frac{\pi\,}{2},\frac{\pi\,}{2}\right)$, if $y\left(x\right)=\int\,\frac{\operatorname{cosec}\,x+\sin\,x}{\operatorname{cosec}\,x\sec\,x+\tan\,x\sin\,^{2}x}dx$ and…

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

Let $f\left(x\right)$ and $g\left(x\right)$ be two functions satisfying $f\left(x^{2}\right)+g(4-x)=4x^{3}$ and $g\left(4-x\right)+g\left(x\right)=0,$ then the value of $\int\,_{-4}^{4}f\left(x^{2}\right)dx$ is

If the normal to the curve$y\left(x\right)=\int\,_{0}^{x}\left(2t^{2}-15t+10\right)dt$ at a point $\left(a,b\right)$ is parallel to the line $x+3y=-5,a>1$, then the value of $\left|a+6b\right|$ is equal to ________.