تمرینات حل شده انتگرال به روش جزء به جزء.
انتگرال گیری به روش جزء به جزء
[math]\int {x\sin xdx} = ? \\ \left\{ \begin{array}{l} u = x \\ dv = \sin xdx \\ \end{array} \right\} \Rightarrow \left\{ \begin{array}{l} du = dx \\ v = – \cos x \\ \end{array} \right\} \\ \int {x\sin xdx} = x( – \cos x) – \int {( – \cos x)dx} \\ = – x\cos x + \int {\cos xdx} \\ = – x\cos x + \sin x + c \\ \\[/math]
سوال 2:
[math] \int x {e^{3x}}dx = ? \\ \left\{ \begin{array}{l} u = x \\ dv = {e^{3x}}dx \\ \end{array} \right\} \Rightarrow \left\{ \begin{array}{l} du = dx \\ v = \frac{1}{3}{e^{3x}} \\ \end{array} \right\} \\ uv – \int {vdu = \frac{{x{e^{3x}}}}{3}} – \int {\frac{{{e^{3x}}}}{3}} dx = \frac{{x{e^{3x}}}}{3} – \frac{{{e^{3x}}}}{9} + c \\[/math]
سوال 3:
[math] \int {x\sin x\cos xdx = ?} \\ \left\{ \begin{array}{l} u = x \\ dv = \sin x\cos xdx \\ \end{array} \right\} \Rightarrow \left\{ \begin{array}{l} du = dx \\ v = \frac{1}{2}{\sin ^2}x \\ \end{array} \right\} \\ uv – \int {vdu = x} (\frac{1}{2}{\sin ^2}x) – \int {\frac{1}{2}{{\sin }^2}xdx} \\ = x(\frac{1}{2}{\sin ^2}x) – \frac{1}{2}\int {{{\sin }^2}xdx} \\ \left[ {{{\sin }^2}x = (\frac{1}{2})(1 – \cos 2x)} \right] \\ = \frac{1}{2}x{\sin ^2}x – \frac{1}{2}\int {(\frac{1}{2})(1 – \cos 2x)dx} \\ = \frac{1}{2}x{\sin ^2}x – \frac{1}{4}\int {(1 – \cos 2x)dx} \\ = \frac{1}{2}x{\sin ^2}x – \frac{1}{4}(x – \frac{{\sin 2x}}{2}) + c \\[/math]
سوال 4:
[math] \int {x\sqrt {x + 3} } dx = ? \\ \left\{ \begin{array}{l} u = x \\ dv = \sqrt {x + 3} dx = {(x + 3)^{\frac{{}}{2}}}dx \\ \end{array} \right\} \Rightarrow \\ \left\{ \begin{array}{l} du = dx \\ v\frac{{(x + 3)}}{{\frac{3}{2}}} = \frac{2}{3}{(x + 3)^{\frac{3}{2}}} \\ \end{array} \right\} \\ uv – \int {vdu = x(\frac{2}{3}} ){(x + 3)^{\frac{3}{2}}} – \int {\frac{2}{3}} {(x + 3)^{\frac{3}{2}}}dx \\ = \frac{2}{3}x{(x + 3)^{\frac{3}{2}}} – \frac{2}{3}\int {{{(x + 3)}^{\frac{3}{2}}}dx} \\ = \frac{2}{3}x{(x + 3)^{\frac{3}{2}}} – \frac{2}{3}\frac{{{{(x + 3)}^{\frac{5}{2}}}}}{{\frac{5}{2}}} + c \\[/math]
سلام
خسته نباشید دستتون درد نکنه فقط کاشکی ی کم تو مثال ها بیشتر توضیح میدادین