heureka

Integral x times sqrt (0.25 - xsquared) dx

$$\small{\text{$ \begin{array}{lrcl} & \int{x\sqrt{\frac{1}{4}-x^2 }} \ dx &=& \frac{1}{2}\int{x \sqrt{1-(2x)^2} \ dx } \\\\ \mathrm{We~ substitute:~} & 2x &=& \sin{(z)} \\ &2\ dx &=& \cos{(z)}\ dz \\\\ & \int{x\sqrt{\frac{1}{4}-x^2 }} \ dx &=& \frac{1}{2}\int{ \frac{\sin{(z)} }{2} \sqrt{1-[\sin{(z)} ]^2} \cdot \frac{ \cos{(z)} }{2}\ dz } \\\\ & \int{x\sqrt{\frac{1}{4}-x^2 }} \ dx &=& \frac{1}{8} \int{ \sin{(z)} \cdot \cos{(z)} \cdot \cos{(z)} \ dz }\\\\ & \int{x\sqrt{\frac{1}{4}-x^2 }} \ dx &=& \frac{1}{8} \int{ \sin{(z)} \cdot \cos^2{(z)}\ dz }\\\\ \end{array} $}}\\\\ \boxed{ \begin{array}{lrcl} \mathrm{Formula:~} & y &=& \frac{ \cos^{n+1}{(z)} } {n+1} \\ \\ & y' &=& \frac{ (n+1)\cdot \cos^{n+1-1}{(z)} \cdot [-\sin{(z)}] }{n+1} \\\\ & y' &=& -\sin{(z)}\cdot \cos^{n}{(z)} \\\\ \mathrm{so:~} & -\int{ \sin{(z)}\cdot \cos^{n}{(z)} \ dz } &=& \frac{ \cos^{n+1}{(z)} } {n+1} \\\\ & \mathbf{\int{ \sin{(z)}\cdot \cos^{n}{(z)} \ dz } } &\mathbf{=}& \mathbf{ -\frac{ \cos^{n+1}{(z)} } {n+1} } \\\\ & \mathbf{\int{ \sin{(z)}\cdot \cos^{2}{(z)} \ dz } } &\mathbf{=}& \mathbf{ -\frac{ \cos^{3}{(z)} } {3} } \end{array} }$$

$$\small{\text{$ \begin{array}{lrcl} & \int{x\sqrt{\frac{1}{4}-x^2 }} \ dx &=& \frac{1}{8} \int{ \sin{(z)} \cdot \cos^2{(z)}\ dz }\\\\ & \int{x\sqrt{\frac{1}{4}-x^2 }} \ dx &=& \frac{1}{8} \left[-\frac{ \cos^{3}{(z)} } {3} \right] + c \\\\ & \int{x\sqrt{\frac{1}{4}-x^2 }} \ dx &=& -\frac{1}{24} \cdot \cos^{3}{(z)} + c \qquad | \qquad \cos{(z)} = \sqrt{1-(2x)^2}\\\\ & \int{x\sqrt{\frac{1}{4}-x^2 }} \ dx &=& -\frac{1}{24} \cdot \left(\sqrt{1-(2x)^2} \right)^3 + c\\\\ & \int{x\sqrt{\frac{1}{4}-x^2 }} \ dx &=& -\frac{1}{24} \cdot \left(1-4x^2\right)^{\frac32} + c \end{array} $}}$$

G(jw)=s-0.5/(s+1)

$$\small{\text{$ \begin{array}{rcl} G(j\omega) &=& j\omega-\dfrac{0.5}{j\omega+1}\\\\ G(j\omega) &=& j\omega-\dfrac{0.5}{1+j\omega}\cdot \left( \dfrac{1-j\omega}{1-j\omega} \right) \\\\ G(j\omega) &=& j\omega-\dfrac{0.5(1-j\omega)}{1-j^2\omega^2} \quad | \quad j^2 = -1\\\\ G(j\omega) &=& j\omega-\dfrac{0.5-0.5 j\omega}{1+\omega^2}\\\\ G(j\omega) &=& j\omega+\dfrac{-0.5+0.5 j\omega}{1+\omega^2}\\\\ G(j\omega) &=& j\omega+ \dfrac{0.5 j\omega}{1+\omega^2} -\dfrac{0.5}{1+\omega^2}\\\\ G(j\omega) &=& -\dfrac{0.5}{1+\omega^2} + \left( \omega+ \dfrac{0.5 \omega}{1+\omega^2} \right) j \\\\ \end{array} $}}$$

GRF 3. Grades: f(x)

H(-1|8)

Gerade: g(x)=-4x+4 berührt f(x) bei x=1

$$\small{\text{$ \begin{array}{l|rclr} \mathbf{ \mathrm{GRF~}3.\mathrm{Grades:~} } & \mathbf{f(x)} & \mathbf{=} & \mathbf{ ax^3+bx^2+cx+d }\\ & \mathbf{f'(x)} & \mathbf{=} & \mathbf{ 3ax^2+2bx+c }\\ \\ \hline \\ \mathrm{H(-1|8)} \quad f(-1)=8 & a(-1)^3+b(-1)^2+c(-1)+d &=& 8\\ & -a+b-c+d &=& 8 & (1)\\ &\\ \mathrm{H(-1|8)} \quad f'(-1)=0 & 3a(-1)^2+2b(-1)+c &=& 0\\ & 3a-2b+c &=& 0 & (2)\\ \\ \hline \\ g(1)=0 \quad f(1)=0 & a(1)^3+b(1)^2+c(1) +d &=& 0\\ & a+b+c+d &=& 0 & (3)\\ &\\ g'(1)=-4 \quad f'(1)=-4 & 3a(1)^2+2b(1)+c &=& -4\\ & 3a+2b+c &=& -4 & (4)\\ \\ \hline \\ (1) & -a+b-c+d &=& 8 \\ (2) & 3a-2b+c &=& 0 \\ (3) & a+b+c+d &=& 0 \\ (4) & 3a+2b+c &=& -4 \\ \\ \hline \\ (4)-(2) & 3a-3a+2b+2b+c-c &=& -4\\ & 4b &=& - 4 \\ & \mathbf{b} & \mathbf{=} & \mathbf{-1} \\ \\ \hline \\ (3)+(1) & a-a+b+b+c-c+d+d &=& 8\\ & 2b+2d &=& 8 \quad | \quad :2 \\ & b+d &=& 4 \\ & (-1)+d &=& 4 \\ & d &=& 4+1 \\ & \mathbf{d} & \mathbf{=} & \mathbf{5} \\ \\ \hline \\ (2)-(3) & 3a-a-2b-b+c-c-d &=& 0\\ & 2a-3b-d &=& 0 \\ & 2a-3(-1)-5 &=& 0 \\ & 2a+3-5 &=& 0 \\ & 2a-2 &=& 0 \\ & 2a &=& 2 \quad | \quad :2 \\ & \mathbf{a} & \mathbf{=} & \mathbf{1} \\ \\ \hline \\ (3) & c &=& -a-b-d\\ & c &=& -1-(-1)-5 \\ & c &=& -1+1-5\\ & \mathbf{c} & \mathbf{=} & \mathbf{-5} \\ \end{array} $}}$$

$$\mathbf{ \mathrm{GRF~}3.\mathrm{Grades:~} } \mathbf{f(x)= x^3-x^2-5x+5 }$$

$$\small{\text{$ \mathbf{ a_{22} = -49 \quad \mathrm{and} \quad a_{25}= -58 \quad \mathrm{find} \quad a_1 } $}}$$

$$\mathbf{ \mathrm{Explicit~ Formula:} & \quad a_n = a_1 + (n-1)\cdot d } \\\\ \small{\text{$ \begin{array}{lrcl} & a_{22} &=& a_1 + 21 \cdot d \\ & a_{25} &=& a_1 + 24 \cdot d \\ \\ \hline \\ & a_{22} - a_1 &=& 21 \cdot d \\ & a_{25} - a_1 &=& 24 \cdot d \\ \\ \hline \\ & \dfrac{ a_{22} - a_1 } { a_{25} - a_1 } &=& \dfrac{21} {24}\\ \\ & \dfrac{ a_{22} - a_1 } { a_{25} - a_1 } &=& \dfrac{7} {8}\\\\ & 8\cdot ( a_{22} - a_1 ) &=& 7\cdot ( a_{25} - a_1 )\\ & 8\cdot a_{22} - 8\cdot a_1 &=& 7\cdot a_{25} - 7\cdot a_1\\ & 8\cdot a_1 - 7\cdot a_1 &=& 8\cdot a_{22} - 7\cdot a_{25} \\ & \mathbf{a_1} &\mathbf{=}& \mathbf{8\cdot a_{22} - 7\cdot a_{25}} \\\\ & a_1 &=& 8\cdot (-49) - 7 \cdot (-58 ) \\ & a_1 &=& -392 + 406 \\ & \mathbf{a_1} &\mathbf{=}& \mathbf{14} \end{array} $}}$$

-2a^4b(-3ab^5(5a^-3b^-7) ) simplify

$$\small$\text{$ \begin{array}{rcl} -2a^4\cdot b \left [-3\cdot ab^5 ( 5a^{-3}\cdot b^{-7} ) \right] \\\\ &=& -2a^4\cdot b \left (-3\cdot 5 \cdot a\cdot a^{-3}\cdot b^5 \cdot b^{-7} \right) \\\\ &=& -2a^4\cdot b \left (-15\cdot a^{1-3}\cdot b^{5-7} \right) \\\\ &=& -2a^4\cdot b \left (-15\cdot a^{-2}\cdot b^{-2} \right) \\\\ &=& (-2)\cdot(-15)\cdot a^4\cdot b \cdot a^{-2}\cdot b^{-2} \right) \\\\ &=& 30\cdot a^4\cdot b \cdot a^{-2}\cdot b^{-2} \\\\ &=& 30\cdot a^4\cdot a^{-2} \cdot b \cdot b^{-2} \\\\ &=& 30\cdot a^{4-2} \cdot b^{1-2} \\\\ &=& 30\cdot a^{2} \cdot b^{-1} \end{array} $}}$$

Hallo radix,

ich bedanke mich recht herzlich.

34° 1D.176    und    09° 17.24B

Umwandlung von Hex nach Dezimal:

$$\small{\text{$ \begin{array}{rcl} 34_{(16)} = 52_{(10)}\\\\ 1D.176_{(16)} = 29,09130859375_{(10)} \\\\ \mathbf{ 34\ensurement{^{\circ}} ~1D.176_{(16)} = 52\ensurement{^{\circ}} ~29,09130859375_{(10)} } \\ \\ \hline \\ \end{array} $}}\\\\ \small{\text{$ \begin{array}{rcl} 09_{(16)} = 09_{(10)}\\\\ 17.24B_{(16)} = 23,143310546875_{(10)} \\\\ \mathbf{ 09\ensurement{^{\circ}} ~17.24B_{(16)} = 09\ensurement{^{\circ}} ~23,143310546875_{(10)} } \end{array} $}}\\\\$$

6a + b = 16
5a - 2b = 19

$$\small\text{$ \begin{array}{lrcl} (1) & 6a + b &=& 16 \qquad | \qquad -6a\\ & \mathbf{b} &=& \mathbf{16 -6a} \\ \\ \hline \\ (2) & 5a - 2b &=& 19 \qquad | \qquad \mathbf{b} = \mathbf{16 -6a}\\ & 5a -2(\mathbf{16 -6a}) &=& 19 \\ & 5a -32 + 12a &=& 19 \\ & 17a -32 &=& 19 \qquad | \qquad +32\\ & 17a &=& 19 +32\\ & 17a &=& 51 \qquad | \qquad :17\\ & a &=& \dfrac{51}{17} \\ & \mathbf{a} & \mathbf{=} & \mathbf{3} \\ \\ \hline \\ & b &=& 16- 6a \qquad | \qquad \mathbf{a} = \mathbf{3} \\ & b &=& 16 - 6 \cdot \mathbf{3} \\ & b &=& 16 - 18 \\ & \mathbf{b} &\mathbf{=} & \mathbf{-2} \\ \\ \hline \\ \end{array} $}}$$

  $$\small{\text{$\dfrac{1265}{12}=105~\dfrac{5}{12} = 105.41\overline{6}$}}$$

1.05 amps connected to 110 volts = ? resistance

$$R_{\Omega}=\dfrac{110 ~volts }{1.05~ amps} = 104.7619 ~ohms$$