If $\int \frac{\left(\sqrt{1+x^2}+x\right)^{10}}{\left(\sqrt{1+x^2}-x\right)^9} \mathrm{~d} x=\frac{1}{\mathrm{~m}}\left(\left(\sqrt{1+x^2}+x\right)^{\mathrm{n}}\left(\mathrm{n} \sqrt{1+x^2}-x\right)\right)+\mathrm{C}$ where C is the constant of integration and $\mathrm{m}, \mathrm{n} \in \mathbf{N}$, then $\mathrm{m}+\mathrm{n}$ is equal to _________ .
If $\int \frac{2 x^2+5 x+9}{\sqrt{x^2+x+1}} \mathrm{~d} x=x \sqrt{x^2+x+1}+\alpha \sqrt{x^2+x+1}+\beta \log _{\mathrm{e}}\left|x+\frac{1}{2}+\sqrt{x^2+x+1}\right|+\mathrm{C}$, where $C$ is the constant of integration, then $\alpha+2 \beta$ is equal to __________ .
If $$\int \frac{1}{\sqrt[5]{(x-1)^4(x+3)^6}} \mathrm{~d} x=\mathrm{A}\left(\frac{\alpha x-1}{\beta x+3}\right)^B+\mathrm{C}$$, where $$\mathrm{C}$$ is the constant of integration, then the value of $$\alpha+\beta+20 \mathrm{AB}$$ is _________.
If $$\int \operatorname{cosec}^5 x d x=\alpha \cot x \operatorname{cosec} x\left(\operatorname{cosec}^2 x+\frac{3}{2}\right)+\beta \log _x\left|\tan \frac{x}{2}\right|+\mathrm{C}$$ where $$\alpha, \beta \in \mathbb{R}$$ and $$\mathrm{C}$$ is the constant of integration, then the value of $$8(\alpha+\beta)$$ equals _________.
and $f(1)=\frac{1}{\alpha \beta} \tan ^{-1}\left(\frac{\alpha}{\beta}\right)$, $\alpha, \beta>0$, then $\alpha^{2}+\beta^{2}$ is equal to ____________.
Let $$I(x)=\int \sqrt{\frac{x+7}{x}} \mathrm{~d} x$$ and $$I(9)=12+7 \log _{e} 7$$. If $$I(1)=\alpha+7 \log _{e}(1+2 \sqrt{2})$$, then $$\alpha^{4}$$ is equal to _________.
$$\alpha {\log _e}|1 + \tan x| + \beta {\log _e}|1 - \tan x + {\tan ^2}x| + \gamma {\tan ^{ - 1}}\left( {{{2\tan x - 1} \over {\sqrt 3 }}} \right) + C$$, when C is constant of integration, then the value of $$18(\alpha + \beta + {\gamma ^2})$$ is ______________.
$$\int {{{({x^2} - 1) + {{\tan }^{ - 1}}\left( {{{{x^2} + 1} \over x}} \right)} \over {({x^4} + 3{x^2} + 1){{\tan }^{ - 1}}\left( {{{{x^2} + 1} \over x}} \right)}}dx} $$
$$ = \alpha {\log _e}\left( {{{\tan }^{ - 1}}\left( {{{{x^2} + 1} \over x}} \right)} \right) + \beta {\tan ^{ - 1}}\left( {{{\gamma ({x^2} + 1)} \over x}} \right) + \delta {\tan ^{ - 1}}\left( {{{{x^2} + 1} \over x}} \right) + C$$
where C is an arbitrary constant, then the value of 10($$\alpha$$ + $$\beta$$$$\gamma$$ + $$\delta$$) is equal to ______________.
$$ \text { Let } f(x)=\int x^3 \sqrt{3-x^2} d x \text {. If } 5 f(\sqrt{2})=-4 \text {, then } f(1) \text { is equal to } $$
Let $\int x^3 \sin x \mathrm{~d} x=g(x)+C$, where $C$ is the constant of integration. If $8\left(g\left(\frac{\pi}{2}\right)+g^{\prime}\left(\frac{\pi}{2}\right)\right)=\alpha \pi^3+\beta \pi^2+\gamma, \alpha, \beta, \gamma \in Z$, then $\alpha+\beta-\gamma$ equals :
Let $\mathrm{I}(x)=\int \frac{d x}{(x-11)^{\frac{11}{13}}(x+15)^{\frac{15}{13}}}$. If $\mathrm{I}(37)-\mathrm{I}(24)=\frac{1}{4}\left(\frac{1}{\mathrm{~b}^{\frac{1}{13}}}-\frac{1}{\mathrm{c}^{\frac{1}{13}}}\right), \mathrm{b}, \mathrm{c} \in \mathcal{N}$, then $3(\mathrm{~b}+\mathrm{c})$ is equal to
If $\int \mathrm{e}^x\left(\frac{x \sin ^{-1} x}{\sqrt{1-x^2}}+\frac{\sin ^{-1} x}{\left(1-x^2\right)^{3 / 2}}+\frac{x}{1-x^2}\right) \mathrm{d} x=\mathrm{g}(x)+\mathrm{C}$, where C is the constant of integration, then $g\left(\frac{1}{2}\right)$ equals :
Let $$\int \frac{2-\tan x}{3+\tan x} \mathrm{~d} x=\frac{1}{2}\left(\alpha x+\log _e|\beta \sin x+\gamma \cos x|\right)+C$$, where $$C$$ is the constant of integration. Then $$\alpha+\frac{\gamma}{\beta}$$ is equal to :
Let $$I(x)=\int \frac{6}{\sin ^2 x(1-\cot x)^2} d x$$. If $$I(0)=3$$, then $$I\left(\frac{\pi}{12}\right)$$ is equal to
If $$\int \frac{1}{\mathrm{a}^2 \sin ^2 x+\mathrm{b}^2 \cos ^2 x} \mathrm{~d} x=\frac{1}{12} \tan ^{-1}(3 \tan x)+$$ constant, then the maximum value of $$\mathrm{a} \sin x+\mathrm{b} \cos x$$, is :
If $$\int \frac{\sin ^{\frac{3}{2}} x+\cos ^{\frac{3}{2}} x}{\sqrt{\sin ^3 x \cos ^3 x \sin (x-\theta)}} d x=A \sqrt{\cos \theta \tan x-\sin \theta}+B \sqrt{\cos \theta-\sin \theta \cot x}+C$$, where $$C$$ is the integration constant, then $$A B$$ is equal to
For $$x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$$, if $$y(x)=\int \frac{\operatorname{cosec} x+\sin x}{\operatorname{cosec} x \sec x+\tan x \sin ^2 x} d x$$, and $$\lim _\limits{x \rightarrow\left(\frac{\pi}{2}\right)^{-}} y(x)=0$$ then $$y\left(\frac{\pi}{4}\right)$$ is equal to
$$\text { The integral } \int \frac{\left(x^8-x^2\right) \mathrm{d} x}{\left(x^{12}+3 x^6+1\right) \tan ^{-1}\left(x^3+\frac{1}{x^3}\right)} \text { is equal to : }$$
For $$\alpha, \beta, \gamma, \delta \in \mathbb{N}$$, if $$\int\left(\left(\frac{x}{e}\right)^{2 x}+\left(\frac{e}{x}\right)^{2 x}\right) \log _{e} x d x=\frac{1}{\alpha}\left(\frac{x}{e}\right)^{\beta x}-\frac{1}{\gamma}\left(\frac{e}{x}\right)^{\delta x}+C$$ , where $$e=\sum_\limits{n=0}^{\infty} \frac{1}{n !}$$ and $$\mathrm{C}$$ is constant of integration, then $$\alpha+2 \beta+3 \gamma-4 \delta$$ is equal to :
If $$I(x) = \int {{e^{{{\sin }^2}x}}(\cos x\sin 2x - \sin x)dx} $$ and $$I(0) = 1$$, then $$I\left( {{\pi \over 3}} \right)$$ is equal to :
The integral $$ \int\left[\left(\frac{x}{2}\right)^x+\left(\frac{2}{x}\right)^x\right] \ln \left(\frac{e x}{2}\right) d x $$ is equal to :
Let $$I(x)=\int \frac{(x+1)}{x\left(1+x e^{x}\right)^{2}} d x, x > 0$$. If $$\lim_\limits{x \rightarrow \infty} I(x)=0$$, then $$I(1)$$ is equal to :
Let $$I(x)=\int \frac{x^{2}\left(x \sec ^{2} x+\tan x\right)}{(x \tan x+1)^{2}} d x$$. If $$I(0)=0$$, then $$I\left(\frac{\pi}{4}\right)$$ is equal to :
Let $$f(x) = \int {{{2x} \over {({x^2} + 1)({x^2} + 3)}}dx} $$. If $$f(3) = {1 \over 2}({\log _e}5 - {\log _e}6)$$, then $$f(4)$$ is equal to
For $$I(x)=\int \frac{\sec ^{2} x-2022}{\sin ^{2022} x} d x$$, if $$I\left(\frac{\pi}{4}\right)=2^{1011}$$, then
$$ \text { The integral } \int \frac{\left(1-\frac{1}{\sqrt{3}}\right)(\cos x-\sin x)}{\left(1+\frac{2}{\sqrt{3}} \sin 2 x\right)} d x \text { is equal to } $$
If $$\int {{1 \over x}\sqrt {{{1 - x} \over {1 + x}}} dx = g(x) + c} $$, $$g(1) = 0$$, then $$g\left( {{1 \over 2}} \right)$$ is equal to :
$$\int {{{\sin \theta .\sin 2\theta ({{\sin }^6}\theta + {{\sin }^4}\theta + {{\sin }^2}\theta )\sqrt {2{{\sin }^4}\theta + 3{{\sin }^2}\theta + 6} } \over {1 - \cos 2\theta }}} \,d\theta $$ is :
$$\int {{{\cos \theta } \over {5 + 7\sin \theta - 2{{\cos }^2}\theta }}} d\theta $$ = A$${\log _e}\left| {B\left( \theta \right)} \right| + C$$,
where C is a constant of integration, then $${{{B\left( \theta \right)} \over A}}$$
can be :
$$\int {\left( {{e^{2x}} + 2{e^x} - {e^{ - x}} - 1} \right){e^{\left( {{e^x} + {e^{ - x}}} \right)}}dx} $$ = $$g\left( x \right){e^{\left( {{e^x} + {e^{ - x}}} \right)}} + c$$
where c is a constant of integration, then g(0) is equal to :
(where C is a constant of integration):
where C is a constant of integration, then the ordered pair (A(x), B(x)) can be :
where C is a constant of integration, then the ordered pair ($$\lambda $$, ƒ($$\theta $$)) is equal to :
(where C is a constant of integration)
and ƒ(0) = 0, then ƒ(1) is equal to :
where c is a constant of integration, then $$\lambda f\left( {{\pi \over 3}} \right)$$ is equal to
$$\int {{{\tan x + \tan \alpha } \over {\tan x - \tan \alpha }}} dx$$ = A(x) cos 2$$\alpha $$ + B(x) sin 2$$\alpha $$ + C, where C is a
constant of integration, then the functions A(x) and B(x) are respectively :
(Here C is a constant of integration)
where C is a constant of integration then :
= esecxf(x) + C then a possible choice of f(x) is :-
where C is a constant of integration, then the function ƒ(x) is equal to
(where c is a constant of integration)
$$f\left( 0 \right) = 0,$$ then the value of $$f(1)$$ is :
$$\int {x\sqrt {{{2\sin ({x^2} - 1) - \sin 2({x^2} - 1)} \over {2\sin ({x^2} - 1) + \sin 2({x^2} - 1)}}} dx} $$ is equal to :
(where c is a constant of integration)
$$\int {{{{{\sin }^2}x{{\cos }^2}x} \over {{{\left( {{{\sin }^5}x + {{\cos }^3}x{{\sin }^2}x + {{\sin }^3}x{{\cos }^2}x + {{\cos }^5}x} \right)}^2}}}} dx$$
is equal to
(where C is a constant of integration), then the ordered pair (A, B) is equal to :
(where C is a constant of integration)
$$\int {} $$f(x) dx = A log$$\left| {} \right.$$1 $$-$$ x $$\left| {} \right.$$ + Bx + C,
then the ordered pair (A, B) is equal to :
(where C is a constant of integration)
$$\int {\sqrt {1 + 2\cot x(\cos ecx + \cot x)\,} \,\,dx} $$
$$\left( {0 < x < {\pi \over 2}} \right)$$ is equal to :
(where C is a constant of integration)
If $${I_4} + {I_6}$$ = $$a{\tan ^5}x + b{x^5} + C$$, where C is a constant of integration,
then the ordered pair $$\left( {a,b} \right)$$ is equal to
(where C is a constant of integration.)
where k is a constant of integration, then A + B +C equals :
equal to :
$$(A, B)$$ is