Algebra, zadanie nr 1969
ostatnie wiadomo艣ci | regulamin | latex
| Autor | Zadanie / Rozwi膮zanie |
thorn post贸w: 1 |  2014-01-25 17:01:51witam potrzebuje rozwi膮za艅 nast臋puj膮cych ca艂ek: 1.\int_ cos^{2}x dx 2.\int_ x e^{2}x dx 3.\int_ x^{2}e^{2} dx 4.\int_ x sin^{2}x dx 5.\int_ ln x dx 6.\int_x ln x dx 7.\int_ ln^{2}x dx |
abcdefgh post贸w: 1255 | 2014-01-25 20:24:50 1. $cos2x=cos^2-sin^2x$ $cos2x=cos^2x-1+cos^2x$ $\frac{1}{2}(cos2x+1)=cos^2x$ $\int \frac{1}{2}(cos2x+1)dx=\frac{1}{2} \int cos2xdx+\frac{1}{2}\int 1dx=\frac{1}{4}sin2x+\frac{1}{2} x+c $ 2. $\int xe^{2}dx=\begin{bmatrix} f(x)=x \ g\'(x)=e^{2x} \\ f\'(x)=1 \ g(x)=\frac{1}{2}e^{2x} \end{bmatrix}=\frac{x}{2}e^{2x}-\frac{1}{2} \int e^{2x}dx= \frac{x}{2}e^{2x}-\frac{1}{4}e^{2x}+c$ |
abcdefgh post贸w: 1255 | 2014-01-25 20:38:13 $\int x^{2}e^{2} dx=e^{2} \int x^2dx=e^{2}*\frac{x^3}{3}+c$ $\int x sin^{2}x dx=\begin{bmatrix} f(x)=x \ g\'(x)=sin^2x \\ f\'(x)=1 \ g(x)=\frac{-1}{2}cos2x \end{bmatrix}=\frac{-x}{2}cos1x+\frac{1}{2}\int cos2xdx=\frac{-x}{2}cos1x+\frac{1}{4}sin2x+c$ $\int ln x dx=\begin{bmatrix} f(x)=lnx \ g\'(x)=1 \\ f\'(x)=\frac{1}{x} \ g(x)=x \end{bmatrix}=xlnx-\int 1dx=xlnx-x$ |
abcdefgh post贸w: 1255 | 2014-01-25 20:42:12 $\int x ln x dx=\begin{bmatrix} f(x)=x \ g\'(x)=lnx \\ f\'(x)=1 \ g(x)=xlnx-x \end{bmatrix}=x^2lnx-x^2-\int xlnx dx + \int x dx=$ $x^2lnx-x^2-\int xlnx dx+\frac{x^2}{2}$ $\int x ln x dx=x^2lnx-x^2-\int xlnx dx+\frac{x^2}{2}$ $2\int x ln x dx=x^2lnx-x^2-\frac{x^2}{2}$ $\int x ln x dx=\frac{1}{2}(x^2lnx-x^2-\frac{x^2}{2})$ |
abcdefgh post贸w: 1255 | 2014-01-25 20:46:44 $\int ln^{2}x dx =\begin{bmatrix} f(x)=ln^2x \ g\'(x)=1 \\ f\'(x)=\frac{2lnx}{x} \ g(x)=x \end{bmatrix}=xln^2x-2\int lnxdx=xln^2x-2(xlnx-x)+c$ |
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