2 2
X\X2 \ X-yXry \ X%i
X\X% ~r X-yXo ~r X2-, (-L)
ry ry ry
JL\Jb2'L?j-
\begin{gather}
x_{l} x_{2} + x_{ir{2} x_{2K{2} + x_{3},\notag\\
x_{l} x_{3} + x_{lK{2} x_{3K{2} + x_{2},\label{E:ml9}\\|
x_{l} x_{2} x_{3}. \notag
\end{gather}
l ry ry ry ry ry I ry ry ry ry ry I ry ry ry ry ry I ry ry ry ry ry \ I
i X1X2X3X4X5 -\- 0,1X3X4X5X5 ~r x^X2X4X5X5 ~r XxX2XßX5Xg 1 ~r
(xix2x3x4 + X1X2X3X5 + X1X2X4X5 + X1X3X4X5 + X2X3X4X5)2
(2)
\ b e g i n { m u l t l i n e } \ l a b e l { E : m l l 3 }
(x_{l} x_{2} x_{3} x_{4} x_{5} x_{6})^{2} + \ \
(x_{l} x_{2} x_{3} x_{4} x_{5} + x_{l} x_{3} x_{4}
x_{5} x_{6} + x_{l} x_{2} x_{4} x_{5} x_{6}
+ x_{l} x_{2} x_{3} x_{5} x _ { 6 } r { 2 } + \ \
(x_{l} x_{2} x_{3} x_{4} + x_{l} x_{2} x_{3} x_{5} +
x_{l} x_{2} x_{4} x_{5} + x_{l} x_{3} x_{4} x_{5} +
x_{2} x_{3} x_{4} x_{5})~{2}
\ e n d { m u l t l i n e }
x = y +z,
u = v + w.
\begin{align} \label{E:mll}
x &= y + z,\\
u &= v + w.\label{E:mlla}
\end{align}
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hix
}{x) + g{x) 1 + }{x)g{x)
1+ P{x V^ sinx
1 + g[x
dx -- 2 tan" x 2
dx
\begin{align} \label{E
h(x) &= \int \left(
ml2}
\frac{f(x) + g(x)Hl + f~{2}(x)}+
\frac{ 1 + f(x)g(x) }{ \sqrt{l - \sin x} }
\right)\,\dx\\
&= \int \frac{ 1 + f(x) }{ 1 + g(x) }\,\dx
-2 \tan~{-l}(x - 2) \notag
\end{align}
x = x A (y V z) (by distributivity) (6)
= (x A y) V (x A z) (by condition (M))
= 2/V z.
\begin{align}
x &= x \ wedge (y \vee z) k &\text{(by distributiv
k= (x \wedge y) \vee (x \wedge z) k &\text{(by condition (
k= y \vee z. \notag
\end{align}
f(x) = x + yz g(x) = x + y + z (7)
h(x) = xy + xz + yz k(x) = (x + y)(x + z) (2/ + 2)
\begin{align}\label{E:mm3}
f(x) &= x + yz & g(x) &= x + y + z\\
h(x) &= xy + xz + yz & k(x) &= (x + y)(x + z)(y + z)
\notag
\end{align}
f(x) = x + yz g(x) = x + y + z (8)
h(x) = xy + xz + yz k(x) = (x + y)(x + z) (2/ + 2)
\begin{flalign}\label{E:mm3f1}
f(x) &= x + yz & g(x) &= x + y + z\\
h(x) &= xy + xz + yz & k(x) &= (x + y)(x + z)(y + z)
\notag
\end{flalign}
x = 17y
y > a+b+c
x = 17y
y > a + b+ c
\begin{eqnarray}
x k = k 17y \ \
y & > & a + b + c
\end{eqnarray}
\begin{align}
x k = 17y \\
y k > a + b + c
\end{align}
X\
\i<5
+ ď
i<5
+ ď
\begin{align}
x_{l} + y_{l} + \left( \sum_{i < 5} \binom{5}{i}
&+ a~{2} \right)^{2}\\
\left( \sum_{i < 5} \binom{5}{i} k+ \alpha~{2} \right)~{2}
\end{align}
X\
\ í<5
í<5
+ ď
+ ď
f(x) = x + yz g (x) = x + y + z (13)
h(x) = xy + xz + yz k (x) = (x + y)(x + z) (2/ + 2)
\begin{alignat}{2} \label{E:ml3}
f(x) &= x + yz & \qquad g(x) k= x + y + z\\
h(x) &= xy + xz + yz k \qquad k(x) k= (x + y)(x + z)
(y + z)\notag
\end{alignat}
x = x A (y V z) , by distributivity,
= (x A y) V (x A z) , by Condition (M),
= yV z
\begin{alignat}{2} \label{E:ml3a}
x k= x \wedge (y \vee z) k &\quad\text{, by
distributivity,}\\
k= (x \wedge y) \vee (x \wedge z)
k &\quad\text{, by
Condition (M),}
\notag\\
k= y \vee z \notag
\end{alignat}
JL -- O.
y = 4,
z = 5;
\begin{aligned}
x k= 3,\\
y &= 4,\\
z &= 5;
\end{aligned}
\text{\qquad or \qquad}
\begin{aligned}
x &= 5,\\
y &= 12,\\
z &= 13.
\end{aligned}
h(x
f(x) + g(x) 1 + f{x)g(x]
1 + P{x y/1 -- sinx
1 + f(x) , T ,
^ ^ dx - 2 tan"1
(x - 2
1 + g(:r
áx
15
\begin{equation}
\begin{aligned} \label{E:longInt2}
h(x) &= \int \left(
[x)g(x) R\s\frac{l+ f(:
rac{f(x) + g(x)Hl+ f~{2}(x)}
qrt{l - \sin x}} \right)\,\dx
&= \int \frac{ 1 + f(x) }{ 1 + g(x) }
\,\dx - 2 \taiT{-l} (x-2)
\end{aligned}
\end{equation}
JL -- O.
y = 4,
z = 5;
\begin{aligned}[b]
x k= 3 , \ \
y &= 4 , \ \
z &= 5;
\end{aligned}
\text{\qquad or \qquad}
\begin{aligned} [b]
x &= 5,\\
y &= 12,\\
z &= 13.
\end{aligned}
í ry ry ry ry ry ry l
-- 1X1X2X3X4X5X5 J
í ry ry ry ry ry I ry ry ry ry ry I ry ry ry ry ry I ry ry ry ry ry \ í \ ľ\ \
-- 1 X1X2X3X4X5 -\- 0,1X3X4X5X5 ~r X^X2X4X5Xg ~r X^X2X3X5Xg 1 v y
= (xiX2X3X4 + X1X2X3X5 + X1X2X4X5 + X1X3X4X5 + X2X3X4X5)2
\begin{equation} \label{E:ml5}
\begin{split}
f &= (x_{l} x_{2} x_{3} x_{4} x_{5} x_{6})^{2}\\
&= (x_{l} x_{2} x_{3} x_{4} x_{5} +
x_{l} x_{3} x_{4} x_{5} x_{6} +
x_{l} x_{2} x_{4} x_{5} x_{6} +
x_{l} x_{2} x_{3} x_{5} x_{6})^{2}\\
&= (x_{l} x_{2} x_{3} x_{4} + x_{l} x_{2} x_{3}
x_{5} + x_{l} x_{2} x_{4} x_{5} + x_{l} x_{3}
x_{4} x_{5} + x_{2} x_{3} x_{4} x_{5})^{2}
\end{split}
\end{equation}
í ry ry ry ry ry ry \
- \X\X2X2,X^X§XQ\
= (X1X2X3X4X5 + X1X3X4X5XQ + XIX2X^X^XQ + X1X2X3X5XQ)2
I f~Y* f~Y* f~Y* f~Y* I ry* ry* ry* /y I ry* ry* f~y* f~Y* I ry* ry* ry* ry* I ry* ry* ry* ry* \
-- I Xx^2^3^4 ~r~ ^ l ^ 2 ^ 3 ^ 5 ~r~ ^ l ^ 2 ^ 4 ^ 5 "I^
l ^ 3 ^ 4 ^ 5 \ d^2^3^A^5) 1
g = 2/12/22/3\begin{align}
\label{E:ml6}
\begin{split}
f k= (x_{l} x_{2} x_{3} x_{4} x_{5} x_{6})^{2}\\
&= (x_{l} x_{2} x_{3} x_{4} x_{5} +
x_{l} x_{3} x_{4} x_{5} x_{6} +
x_{l} x_{2} x_{4} x_{5} x_{6} +
x_{l} x_{2} x_{3} x_{5} x_{6})^{2}\\
&= (x_{l} x_{2} x_{3} x_{4} +
x_{l} x_{2} x_{3} x_{5} +
x_{l} x_{2} x_{4} x_{5} +
x_{l} x_{3} x_{4} x_{5} +
x_{2} x_{3} x_{4} x_{5})^{2},
\end{split}\\
g &= y_{l} y_{2} y_{3}. \label{E:ml7}
\end{align}
hi x
f (x) + g (x) + 1 + f{x)g(x) ( ^
1 + /2
(x VT^
19
srna;
The reader may find the following form easier to read:
/ 1 + /
/
W
dx-2tan-1
X 2
\begin{align} \label{E:ml8}
h(x) &= \int \left(\frac{f(x) + g(x)}{l + f~{2}(x)} +
\frac{l + f (x)g(x)H\sqrt{l - \sin x}}\right)\,\dx\\
\intertext{The reader may find the following
form easier to read:}
k= \int \frac{l + f(x)}{l + g(x)}\,\dx 2
\tan~{-l}(x - 2) \notag
\end{align}
f (x) = x + yz g{x) = x + y + z
The reader also may find the following polynomials useful:
h(x) = xy + xz + yz k(x) = (x + y)(x + z)(?/ + 2)
\begin{alignat*}{2}
f(x) k= x + yz & \qquad g(x) &= x + y + z \\
\intertext{The reader also may find the following
polynomials useful:}
h(x) k= xy + xz + yz
& \qquad k(x) &= (x + y)(x + z)(y + z)
\end{alignat*}
f (x
-x2
, if x < 0;
0 + x, i f 0 < z < l ; 20
x" otherwise.
\begin{equation} \label{E:mllO}
f (x) =
\begin{cases}
-x~{2}, &\text{if $x \leq 0$;}\\
0 + x, &\text{if $ 0 \leq x \leq 1$;}\\
x~{2}, &\text{otherwise.}
\end{cases}
\end{equation}
í f~Y* f~Y* f~Y* f~Y* f~Y* f~Y* \
-- I Xx^2^3^4^5^6y
I f~Y* f~Y* f~Y* f~Y* f~Y* I /y /y /y /y /y I /y /y /y /y /y I /y /y /y /y /y \ ^
-- I Xx^2^3^4^5 ~r~ ^ l ^ 3 ^ 4 ^ 5 ^ 6 "I^
l ^ 2 ^ 4 ^ 5 ^ 6 ~r~ ^ l ^ 2 ^ 3 ^ 5 ^ 6 y
I f~Y* f~Y* f~Y* f~Y* I /y /y /y /y I /y /y /y /y I /y /y /y /y I /y /y /y /y \
-- I Xx^2^3^4 "I^
l ^ 2 ^ 3 ^ 5 "I^
l ^ 2 ^ 4 ^ 5 "I^
l ^ 3 ^ 4 ^ 5 ~r d^2^3^A^5)
9 = 2/12/22/3 (21)
/l = ^ ^ (22)
\begin{gather} \label{E:mlll}
\begin{split}
f k= (x_{l> x_{2} x_{3} x_{4} x_{5} x _ { 6 } r { 2 } \ \
&= (x_{l} x_{2} x_{3} x_{4} x_{5}
+ x_{l} x_{3} x_{4} x_{5} x_{6} +
x_{l} x_{2} x_{4} x_{5} x_{6} +
x_{l} x_{2} x_{3} x_{5} x_{6})^{2}\\
&= (x_{l} x_{2} x_{3} x_{4} +
x_{l} x_{2} x_{3} x_{5} +
x_{l} x_{2} x_{4} x_{5} +
x_{l} x_{3} x_{4} x_{5} +
x_{2} x_{3} x_{4} x_{5})~{2}
\ e n d { s p l i t } \ \
\begin{align*}
g &= y_{l} y_{2} y_{3}\\
h k= z_{lK{2} z_{2K{2} z_{3K{2}
\end{align*}
\end{gather}
a = b + c,
d = e +/,
x = 2/ + z,
w = v + w.
{\allowdisplaybreaks
\begin{align} \label{E:mll4}
\allowdisplaybreaks
a &= b + c,\\
d &= e + f,\\
x &= y + z,\\
u &= v + w.\notag
\end{align}
}