integrals.tex

\begin{tabular}{l|l|l}
    \multicolumn{3}{c}{Определенный интеграл (1)} \\
    \hline
    
    $\displaystyle \int adx = ax + C$ & 
    $\displaystyle \int \sin{x}dx = -\cos{x} + C$ & 
    $\displaystyle \int \frac{dx}{\sqrt{a^2 - x^2}} = \arcsin{\frac{x}{a}} + C = -\arccos{\frac{x}{a}} + C$  \\
    
    $\displaystyle \int x^ndx = \frac{x^{n+1}}{n+1} + C$ & 
    $\displaystyle \int \cos{x}dx = \sin{x} + C$ & 
    $\displaystyle \int \frac{dx}{a^2+x^2} = \frac{1}{a}\arctan{\frac{x}{a}} + C = -\frac{1}{a}arcctg\frac{x}{a} + C$  \\
    
    $\displaystyle \int \frac{dx}{x} = \ln{|x|} + C$ & 
    $\displaystyle \int \frac{dx}{\sin^2{x}} = -ctg x + C$ & 
    $\displaystyle \int\frac{dx}{x^2 - a^2} = \frac{1}{2a}\ln{\left|\frac{x+a}{x-a}\right|} + C$  \\
    
    $\displaystyle \int a^xdx = \frac{a^x}{\ln{a}} + C$ & 
    $\displaystyle \int \frac{dx}{\cos^2{x}} = \tan{x} + C$ & 
    $\displaystyle \int\frac{dx}{\sqrt{x^2 \pm a^2}} = \ln{|x + \sqrt{x^2 \pm a^2}|}$  \\
    
    $\displaystyle \int e^xdx = e^x + C$ & 
    $\displaystyle $ & 
    $\displaystyle \int\frac{ax+b}{cx+d}dx = \frac{a}{c}x + \frac{bc - ad}{c^2}\ln{|cx + d|} + C$  \\
    \hline
\end{tabular}

\vspace{2ex}
\begin{tabular}{l|l}
    \multicolumn{2}{c}{Определенный интеграл (2)} \\
    \hline
    
    $\displaystyle \int\frac{xdx}{a^2 \pm x^2} = \pm\frac{1}{2}\ln{|a^2 \pm x^2|} + C$ &
    $\displaystyle \int\sqrt{a^2 - x^2}dx = \frac{x}{2}\sqrt{a^2 -x^2} + \frac{a^2}{2}\arcsin{\frac{x}{a}} + C, a > 0$ \\
    
    $\displaystyle \int\frac{dx}{(x+a)(x+b)} = \frac{1}{a-b}\ln{\left|\frac{x + b}{x + a}\right|} + C$ &
    $\displaystyle \int\sqrt{x^2 \pm a^2}dx = \frac{x}{2}\sqrt{x^2+a^2} \pm \frac{a^2}{2}\ln{|x + \sqrt{x^2 \pm a^2}|} + C, a > 0 $ \\
    
    $\displaystyle \int\frac{xdx}{(x+a)(x+b)} = \frac{1}{a - b}\ln\frac{|x+a|^a}{|x+b|^b} + C$ &
    $\displaystyle \int\frac{dx}{\sqrt{a^2 - x^2}} = \arcsin{\frac{x}{a}} + C$ \\
    
    $\displaystyle \int\frac{dx}{ax^2+bx+c} = \frac{2\arctan{\frac{2ax+b}{\sqrt{ac-b^2}}}}{\sqrt{4ac-b^2}} + C$ &
    $\displaystyle \int\frac{xdx}{\sqrt{a^2\pm x^2}} = \pm\sqrt{a^2\pm x^2} + C, a>0$ \\
    
    \multicolumn{2}{l}{$\displaystyle \int\frac{Mx+N}{x^2 + px + q}dx = \frac{M}{2}\ln{(x^2+px+q)} + \left(N-\frac{Mp}{2}\right)\frac{1}{\sqrt{q - \frac{p^2}{4}}}\arctan{\frac{x+\frac{p}{2}}{\sqrt{q-\frac{p^2}{4}}}} + C$} \\
    \hline
\end{tabular}

Формула Лейбница: $\displaystyle \frac{d}{dy} I(y) = \frac{d}{dy} \int\limits_{a(y)}^{b(y)}f(x,y)\,dx  = f\big(b(y),y\big)\cdot \frac{d}{dy} b(y) - f\big(a(y),y\big)\cdot \frac{d}{dy} a(y) + \int\limits_{a(y)}^{b(y)}\frac{\partial}{\partial y} f(x,y) \,dx$