∫f′(x) * g(x) = f(x)g(x) − ∫f(x)g′(x) $$\int_{}^{}{\frac{f'(x)}{f(x)} = ln|f(x)|}$$ $$\int_{}^{}{\frac{f'(x)}{\sqrt{f(x)}} = 2\sqrt{f(x)}}$$ $$\int_{}^{}{f^{n}\left( x \right)*f^{'}\left( x \right) = \frac{f^{n + 1}(x)}{n + 1}}$$

$$\int_{}^{}{arcsinx = x*arcsinx + \sqrt{1 - x^{2}}}$$ $$\int_{}^{}{arccosx = x*arcsinx - \sqrt{1 - x^{2}}}$$ $$\int_{}^{}\text{arctgx} = xarctgx - \frac{1}{2}ln|1 + x^{2}|$$ $$\int_{}^{}\text{arcctgx} = xarcctgx + \frac{1}{2}ln|1 + x^{2}|$$ ∫lnx = xlnx − x $$\int_{}^{}\text{tgax} = - \frac{1}{a}ln|cosax|$$ $$\int_{}^{}\text{ctgax} = \frac{1}{a}ln|sinax|$$ $$\int_{}^{}{\frac{1}{ax + b} = \frac{1}{a}ln|ax + b|}$$ $$\int_{}^{}{{(ax + b)}^{n} = \frac{1}{a}*\frac{{(ax + b)}^{n + 1}}{n + 1}}$$ $$\int_{}^{}{\frac{1}{ax^{2} + bx + c} = 2arctg\left( \frac{2ax + b}{\sqrt{4ac - b^{2}}} \right)*\frac{1}{\sqrt{4ac - b^{2}}}}$$ $$\int_{}^{}{\frac{1}{x^{2} + a^{2}} = \frac{1}{a}\text{arctg}\frac{x}{a}}$$ $$\int_{}^{}{\frac{1}{x^{2} - a^{2}} = \frac{1}{2a}ln|\frac{x - a}{x + a}|}$$ $$\int_{}^{}{\frac{1}{x^{2} + a} = \frac{\sqrt{a}}{a}\text{arctg}\frac{x}{\sqrt{a}}}$$ $$\int_{}^{}{\frac{1}{{(x - k)}^{2} + a} =}\frac{\sqrt{a}}{a}\text{arctg}\frac{x - k}{\sqrt{a}}$$ $$\int_{}^{}{\frac{1}{\sqrt{x^{2} + a}} = ln|x + \sqrt{x^{2} + a\ }|}$$ $$\int_{}^{}{\frac{1}{\sqrt{{(x + k)}^{2} + a}} = ln|x + k + \sqrt{{(x + k)}^{2} + a\ }|}$$ $$\int_{}^{}{\frac{1}{\sqrt{a^{2} - x^{2}}} = arcsin\frac{x}{a}}$$ $$\int_{}^{}{\frac{a}{{(x - k)}^{n}} = \frac{a}{1 - n}*\frac{1}{{(x - k)}^{n - 1}}}$$

∫1 = x                |y = a(x − p)^2+q $$\int_{}^{}{x^{n} = \frac{x^{n + 1}}{n + 1}\text{\ \ \ \ \ \ }\mathbf{|p =}\frac{\mathbf{- b}}{\mathbf{2}\mathbf{a}}}\ ,\ \mathbf{q = -}\frac{\mathbf{\hat{}}}{\mathbf{4}\mathbf{a}}$$ ∫X^−1 = ln|x|    |x=logaa^x,x = lne^x $$\int_{}^{}{a^{x} = \frac{a^{x}}{\text{lna}}}\text{\ \ \ \ \ \ \ \ \ \ }\mathbf{\ t = tg}\frac{\mathbf{x}}{\mathbf{2}}\ ,\ \mathbf{sinx =}\frac{\mathbf{2}\mathbf{t}}{\mathbf{1 +}\mathbf{t}^{\mathbf{2}}}$$ $$\int_{}^{}{e^{x} = e^{x}}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ }\mathbf{cossx =}\frac{\mathbf{1 -}\mathbf{t}^{\mathbf{2}}}{\mathbf{1 +}\mathbf{t}^{\mathbf{2}}}$$ $$\int_{}^{}\text{sinax} = - \frac{1}{a}\text{cosx}$$ $$\int_{}^{}{\sin^{2}\text{ax}} = - \frac{1}{2a}(ax - sinaxcosax)$$ $$\int_{}^{}\text{cosax} = \frac{1}{a}\text{sinx}$$ $$\int_{}^{}{\cos^{2}\text{ax}} = \frac{1}{2a}(ax - sinaxcosax)$$ ∫xsinx = −xcosx + sinx $$\int_{}^{}{\frac{1}{\text{sinax}} = \frac{1}{a}ln|tg\frac{\text{ax}}{2}|}$$ $$\int_{}^{}{\frac{1}{\sin^{2}\text{ax}} = - \frac{1}{a}\text{ctgax}}$$ $$\int_{}^{}{\frac{1}{\text{cosax}} = \frac{1}{a}ln|tg\frac{\text{ax}}{2} + \frac{\pi}{4}|}$$ $$\int_{}^{}{\frac{1}{\cos^{2}\text{ax}} = \frac{1}{a}\text{tgax}}$$