• $\begin{array}{l}series\left(\cos \right(x),x,0,5)\\ \mathrm{>>>}1-\frac{x^2}{2}+\frac{x^4}{\mathrm{24}}+O(x^5)\end{array}$

• $\begin{array}{l}\Gamma \left(1\right)\\ \mathrm{>>>}1\end{array}$

• $\begin{array}{l}\zeta \left(2\right)\\ \mathrm{>>>}1.64493406684823\end{array}$

• $\begin{array}{l}\underset{x\to 0}{lim}\frac{\sin \left(x\right)}{x}\\ \mathrm{>>>}1\end{array}$

• $\begin{array}{l}\underset{x\to 0^{+}}{lim}\frac{1}{x}\\ \mathrm{>>>}\mathrm{oo}\end{array}$

• $\begin{array}{l}\underset{x\to 0^{-}}{lim}\frac{1}{x}\\ \mathrm{>>>}\mathrm{-oo}\end{array}$

• $\begin{array}{l}\int \mathrm{xdx}\\ \mathrm{>>>}0.5*x^2\end{array}$

• $\begin{array}{l}{\int }_0^{\infty }{e}^{-x}\mathrm{xdx}\\ \mathrm{>>>}1\end{array}$

Repeat the corresponding integral operation.

• $\begin{array}{l}\frac{d\ln \left(x\right)}{\mathrm{dx}}\\ \mathrm{>>>}{x}^{-1}\end{array}$

• $\begin{array}{l}y=ln\left(x\right)\\ y^{\text{'}}\left(x\right)\\ \mathrm{>>>}{x}^{-1}\end{array}$

• $\begin{array}{l}\frac{\partial y*\ln \left(x\right)}{\mathrm{\partial x}}\\ \mathrm{>>>}\frac{y}{x}\end{array}$

$\begin{array}{l}u\mathrm{(t)}=\left\{\begin{array}{l}1\mathrm{x>0}\\ \frac{1}{2}\mathrm{x=0}\\ 0\mathrm{otherwise}\end{array}\right.\end{array}$

$\begin{array}{l}F\left(k\right)=\mathrm{FT}(f,x,k)={\int }_{-\infty }^{\infty }f\left(x\right)e^{\mathrm{-2\pi ixk}}\mathrm{dx}\end{array}$

• $\begin{array}{l}F=\mathrm{FT}(e^{-x^2},x,k)\\ \mathrm{>>>}F=\sqrt{\pi }*e^{-{\pi }^2*k^2}\end{array}$

$\begin{array}{l}f\left(x\right)={\mathrm{FT}}^{-1}(F,k,x)={\int }_{-\infty }^{\infty }F\left(k\right)e^{\mathrm{2\pi ixk}}\mathrm{dk}\end{array}$

• $\begin{array}{l}F=\mathrm{FT}(e^{-x^2},x,k)\\ {\mathrm{FT}}^{-1}(F,k,x)\\ \mathrm{>>>}{e}^{-x^2}\end{array}$

$\begin{array}{l}F\left(s\right)=\mathrm{LT}(f,t,s)={\int }_0^{\infty }{e}^{-\mathrm{st}}f\left(t\right)\mathrm{dt}\end{array}$

• $\begin{array}{l}F=\mathrm{LT}(1,t,s)\\ \mathrm{>>>}F=\frac{1}{s}\end{array}$

$\begin{array}{l}f\mathrm{(t)}={\mathrm{LT}}^{-1}\mathrm{(F},s,\mathrm{t)}={\int }_{\mathrm{c-i}\infty }^{\mathrm{c+i}\infty }{e}^{-\mathrm{st}}F(s)\mathrm{ds}\end{array}$

• $\begin{array}{l}F=\mathrm{LT}(1,t,s)\\ {\mathrm{LT}}^{-1}(F,s,t)\\ \mathrm{>>>}u\left(t\right)\end{array}$

$\begin{array}{l}F\left(s\right)=\mathrm{MT}(f,t,s)={\int }_0^{\infty }{x}^{\mathrm{s-1}}f\left(x\right)\mathrm{dx}\end{array}$

• $\begin{array}{l}F=\mathrm{MT}(e^{\mathrm{-t}},t,s)\\ \mathrm{>>>}F=\Gamma \left(s\right)\end{array}$

$\begin{array}{l}f(t)={\mathrm{MT}}^{-1}(F,s,t)={\int }_{\mathrm{c-i}\infty }^{\mathrm{c+i}\infty }{x}^{-s}F(s)\mathrm{ds}\end{array}$

• $\begin{array}{l}F=\mathrm{MT}(e^{\mathrm{-t}},t,s)\\ {\mathrm{MT}}^{-1}(F,s,t)\\ \mathrm{>>>}{e}^{\mathrm{-t}}\end{array}$