\int \frac{dx}{a+bx^2} = \frac{1}{\sqrt{ab}} \tan^{-1} (x \sqrt{\frac{b}{a}})


\int \frac{dx}{\sqrt{x^2\pm a^2}} = \log \frac{(x+\sqrt{x^2\pm a^2})}{a}


\int \frac{dx}{\sqrt{2ax-x^2}} = \operatorname{vers}^{-1} \frac{x}{a}


\int \frac{dx}{\sqrt{x^2\pm 2ax}} = \log (x\pm a + \sqrt{x^2\pm 2ax}


\therefore \text{semi-area} = \frac{c (c^2-a^2)}{4 a} sin^{-1} \frac{\sqrt{c^2 - a^2}}{\sqrt{c^2 - a^2}}


\text{area circle} = \frac{a}{2} 2\pi b = \pi ab


= \pi b^2 \left(3 + \frac{2 b}{a}\right)


\therefore -1=3A, \therefore A=-\frac{1}{3}


\tan \theta = \frac{\sin \theta}{\sqrt{1-\sin^2 \theta}} = \frac{x^{\frac{3}{2}}}{\sqrt{a^3-x^3}}


\therefore \int \frac{\sqrt{x} dx}{\sqrt{a^3-x^3}} = \frac{2}{3} \tan^{-1} \sqrt{\frac{x^3}{a^3-x^3}}


\therefore \int \frac{x dx}{\sqrt{2 a x+x^{2}}}


A N'P' = \int \beta d\alpha = \frac{2}{3\sqrt{3} a}\int (\alpha - 2a)^{\frac{3}{2}} d\alpha


\frac{\partial^{2} z}{\partial x^{2}}, \frac{\partial^{2} z}{\partial x \partial y}, \frac{\partial^{2} z}{\partial y^{2}}


f_{x} + \lambda \phi_{x} + \mu \psi_{x} = 0


\phi_{y} + \phi_{z} \frac{\partial z}{\partial y} + \phi_{t} \frac{\partial t}{\partial y} = 0


\psi_{y} + \psi_{z} \frac{\partial z}{\partial y} + \psi_{t} \frac{\partial t}{\partial y} = 0


f_{y} + f{z} \frac{\partial z}{\partial y} + f_{t} \frac{\partial t}{\partial y} = 0


f_{y} + \lambda \phi_{y} + \mu \psi_{y} = 0


f_{z} + \lambda \phi_{z} + \mu \psi_{z} = 0


f_{t} + \lambda \phi_{t} + \mu \psi_{t} = 0


\frac{\partial(\phi, \psi)}{\partial(z, t)} \neq 0


\frac{\partial(\phi, \psi)}{\partial(x, y)}, \frac{\partial(\phi, \psi)}{\partial(x, z)}, \dots, \frac{\partial(\phi, \psi)}{\partial(z, t)}


\text{div} \mathbf{u} = \frac{\partial u_{1}}{\partial x_{1}} + \frac{\partial u_{2}}{\partial x_{2}} + \frac{\partial u_{3}}{\partial x_{3}}


\text{div} \mathbf{u} = \frac{\partial \omega_{1}}{\partial \xi_{1}} + \frac{\partial \omega_{2}}{\partial \xi_{2}} + \frac{\partial \omega_{3}}{\partial \xi_{3}}


\frac{\partial u_{1}}{\partial x_{1}} + \frac{\partial u_{2}}{\partial x_{2}} + \frac{\partial u_{3}}{\partial x_{3}} = \frac{\partial \omega_{1}}{\partial \xi_{1}} + \frac{\partial \omega{2}}{\partial \xi_{2}} + \frac{\partial \omega_{3}}{\partial \xi_{3}}


\omega_1 = \alpha_1u_1 + \beta_1u_2 + \gamma_1u_3


\omega_2 = \alpha_2u_1 + \beta_2u_2 + \gamma_2u_3


\omega_3 = \alpha_3u_1 + \beta_3u_2 + \gamma_3u_3


u_1 = \alpha_1\omega_1 + \alpha_2\omega_2 + \alpha_3\omega_3


u_2 = \beta_1\omega_1 + \beta_2\omega_2 + \beta_3\omega_3


u_3 = \gamma_1\omega_1 + \gamma_2\omega_2 + \gamma_3\omega_3


\lim_{\delta \to 0} \iint_{R} f(x, y) dS = \iint_{R} f(x, y) dS


d cos u = d sin \left(\frac{\pi}{2}-u\right) = cos \left(\frac{\pi}{2}-u\right) d\left(\frac{\pi}{2}-u\right) = - sin u du


d tan u = \frac{cos u d sin u - sin u d cos u}{cos^2 u} = \frac{cos^2 u du + sin^2 u du}{cos^2 u}


\frac{dy}{dx} = sec 2 x \frac{d}{dx} tan 2 x + tan 2 x \frac{d}{dx} sec 2 x


= sec 2 x sec^2 2 x (2) + tan-2 x sec 2 x tan 2 x (2)


\quad \frac{dy}{dx} = 6 (cos 3 x - sin 2 x)


y = v_0 t \sin{\alpha} - \frac{1}{2} gt^2


(sin x + y) dy + (y cos x - x^{2}) dx = 0


\begin{vmatrix} 4244 & 4245 \\ 2 & 2 \end{vmatrix} = \begin{vmatrix} 4244 & 1 \\ 2 & 0 \end{vmatrix} = -2


\angle AOB = \angle BOC = \angle COD = 60\degree


\frac{a-b}{2} \text{ and } \frac{3a+b}{2}


a^m + ma^{m-1} z+m \times \frac{m-1}{2} \times a^{m-2} z \times \sqrt{\frac{N-a^m}{ma^{m-1}}} = N


a^m + ma^{m-1} z +m \times \frac{m-1}{2} \times a^{m-2} z \times \frac{N-a^m}{ma^{m-1}} = N


ma^{m-1} z+m \times \frac{m-1}{2} \times a^{m-2} \times \sqrt{\frac{N-a^m}{ma^{m-1}}} \times z = N-a^m


z \times \sqrt{ma^{m-1} +m \times \frac{m-1}{2} \times a^{m-2} \times \sqrt{\frac{N-a^m}{ma^{m-1}}}} = N-a^m


z (= \frac{N-a^m}{ma^{m-1} +m \times \frac{m-1}{2} \times a^{m-2} \times \sqrt{\frac{N-a^m}{ma^{m-1}}}}


=\frac{2 \times \overline {N-a^m}}{2ma^{m-1} + \overline{m-1} a^{m-2} \times \sqrt{\frac{N-a^m}{a^{m-1}}}}


=\frac{2 \times \overline{N-a^m}}{2ma^{2m-2} + \overline{m-1} \times a^{m-2} \times \overline{N-a^m}}