Substitution

The value of the integral $\displaystyle\int_{\pi/6}^{\pi/3}\left(\frac{4-\operatorname{cosec}^2x}{\cos^4x}\right)dx$ is:

Let $y=y(x)$ be the solution of the differential equation
$$(\tan x)^{1/2}\,dy=\left(\sec^3x-(\tan x)^{3/2}\,y\right)dx,\quad 0<x<\frac\pi2,\quad y\left(\frac\pi4\right)=\frac{6\sqrt2}5.$$
If $y\left(\dfrac\pi3\right)=\dfrac45\alpha$, then $\alpha^4$ equals ________.

Let $y=y(x)$ be the solution of the differential equation $\left(x^2-x\sqrt{x^2-1}\right)dy+\left(y\left(x-\sqrt{x^2-1}\right)-x\right)dx=0$, $x\ge1$. If $y(1)=1$, then the greatest integer less than $y\left(\sqrt5\right)$ is ________.

The value of the integral $\displaystyle\int_0^2\frac{\sqrt{x\left(x^2+x+1\right)}}{\left(\sqrt{x+1}\right)\left(\sqrt{x^4+x^2+1}\right)}\,dx$ is equal to :

Consider the equation
$$\int_1^e \frac{(\log_e x)^{1/2}}{x\left(a - (\log_e x)^{3/2}\right)^2}\,dx = 1, \qquad a \in (-\infty, 0)\cup(1,\infty).$$
Which of the following statements is/are TRUE?

The value of $\displaystyle\int_0^\pi\frac{e^{\cos x}\sin x}{\left(1+\cos^2x\right)\left(e^{\cos x}+e^{-\cos x}\right)}dx$ is equal to :

If $\displaystyle\int\frac1x\sqrt{\frac{1-x}{1+x}}\,dx=g(x)+c$, $g(1)=0$, then $g\left(\dfrac12\right)$ is equal to :

The integral $\dfrac{24}\pi\displaystyle\int_0^{\sqrt2}\frac{\left(2-x^2\right)dx}{\left(2+x^2\right)\sqrt{4+x^4}}$ is equal to ______.

Let $y=y(x)$ be the solution of the differential equation
$$\frac{dy}{dx}+\frac{\sqrt2y}{2\cos^4x-\cos2x}=xe^{\tan^{-1}\left(\sqrt2\cot2x\right)},\quad0<x<\frac\pi2$$
with $y\left(\dfrac\pi4\right)=\dfrac{\pi^2}{32}$. If $y\left(\dfrac\pi3\right)=\dfrac{\pi^2}{18}e^{-\tan^{-1}(\alpha)}$, then the value of $3\alpha^2$ is equal to ______.

Let $f(t)=\displaystyle\int_0^te^{x^3}\left(\frac{x^8}{\left(x^6+2x^3+2\right)^2}\right)dx$. If $f(1)+f'(1)=\alpha e-\dfrac16$, then the value of $150\alpha$ is equal to ______.

The slope of the tangent to a curve $C:y=y(x)$ at any point $(x,y)$ on it is $\dfrac{2e^{2x}-6e^{-x}+9}{2+9e^{-2x}}$. If C passes through the points $\left(0,\dfrac12+\dfrac\pi{2\sqrt2}\right)$ and $\left(\alpha,\dfrac12e^{2\alpha}\right)$, then $e^\alpha$ is equal to :

The integral $\displaystyle\int\frac{\left(1-\frac1{\sqrt3}\right)(\cos x-\sin x)}{\left(1+\frac2{\sqrt3}\sin2x\right)}dx$ is equal to

Let $y=y(x)$ be the solution curve of the differential equation $\sin\left(2x^2\right)\log_e\left(\tan x^2\right)dy+\left(4xy-4\sqrt2x\sin\left(x^2-\dfrac\pi4\right)\right)dx=0$, $0<x<\sqrt{\dfrac\pi2}$, which passes through the point $\left(\sqrt{\dfrac\pi6},1\right)$. Then $\left|y\left(\sqrt{\dfrac\pi3}\right)\right|$ is equal to ______.

If $\displaystyle\int_0^{\sqrt3}\frac{15x^3}{\sqrt{1+x^2+\sqrt{\left(1+x^2\right)^3}}}dx=\alpha\sqrt2+\beta\sqrt3$, where $\alpha$, $\beta$ are integers, then $\alpha+\beta$ is equal to ______.

The integral $\displaystyle\int_0^{\pi/2}\frac1{3+2\sin x+\cos x}\,dx$ is equal to :

If $\displaystyle\int\frac{\cos x-\sin x}{\sqrt{8-\sin2x}}dx=a\sin^{-1}\left(\frac{\sin x+\cos x}b\right)+c$, where c is a constant of integration, then the ordered pair (a, b) is equal to :

If

$$I = \frac2\pi\int_{-\pi/4}^{\pi/4}\frac{dx}{(1 + e^{\sin x})(2 - \cos2x)}$$

then $27I^2$ equals ______

The value of the integral

$$\int_0^{\pi/2}\frac{3\sqrt{\cos\theta}}{\left(\sqrt{\cos\theta} + \sqrt{\sin\theta}\right)^5}\,d\theta$$

equals ______

The value of the integral

$$\int_0^{1/2} \frac{1 + \sqrt{3}}{\left((x+1)^2 (1-x)^6\right)^{1/4}}\, dx$$

is _____ .

If $y = y(x)$ satisfies the differential equation

$$8\sqrt{x}\left(\sqrt{9 + \sqrt{x}}\right)dy = \left(\sqrt{4 + \sqrt{9 + \sqrt{x}}}\right)^{-1}dx, \quad x > 0$$

and $y(0) = \sqrt{7}$, then $y(256) =$