Indefinite Integration

Let $y=y(x)$ be the solution of the differential equation $x\left(1-x^2\right)\dfrac{dy}{dx}+\left(3x^2y-y-4x^3\right)=0$, $x>1$, with $y(2)=-2$. Then $y(3)$ 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 ______.

$\displaystyle\lim_{n\to\infty}\sum_{r=1}^n\frac r{2r^2-7rn+6n^2}$ 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 :

If $n(2n+1)\displaystyle\int_0^1\left(1-x^n\right)^{2n}dx=1177\int_0^1\left(1-x^n\right)^{2n+1}dx$, then $n\in\mathbf N$ 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 ______.

Let $I_n(x)=\displaystyle\int_0^x\frac1{\left(t^2+5\right)^n}dt$, $n=1,2,3,\ldots$. Then :

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

For $I(x)=\displaystyle\int\frac{\sec^2x-2022}{\sin^{2022}x}\,dx$, if $I\left(\dfrac\pi4\right)=2^{1011}$, then

Let $y=y(x)$ be the solution curve of the differential equation $\dfrac{dy}{dx}+\left(\dfrac{2x^2+11x+13}{x^3+6x^2+11x+6}\right)y=\dfrac{(x+3)}{x+1}$, x > −1, which passes through the point (0, 1). Then y(1) is equal to :

The value of $\dfrac{48S_2}{\pi^2}$ is ____.

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 :

Which of the following inequalities is/are TRUE?

Let $f:\mathbb R\to\mathbb R$ be a differentiable function such that its derivative $f'$ is continuous and $f(\pi) = -6$. If $F:[0,\pi]\to\mathbb R$ is defined by $F(x) = \int_0^x f(t)\,dt$, and if

$$\int_0^\pi\left(f'(x) + F(x)\right)\cos x\,dx = 2,$$

then the value of $f(0)$ is ______

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 _____ .