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

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