Mister Exam
Least common denominator exp(x)/(2*(e^x+1))-exp(x)/(2*(e^x-1))
The solution
You have entered
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x x
e e
---------- - ----------
/ x \ / x \
2*\E + 1/ 2*\E - 1/
$$\frac{e^{x}}{2 \left(e^{x} + 1\right)} - \frac{e^{x}}{2 \left(e^{x} - 1\right)}$$
exp(x)/((2*(E^x + 1))) - exp(x)/(2*(E^x - 1))
Fraction decomposition
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-1/(2*(1 + exp(x))) - 1/(2*(-1 + exp(x)))
$$- \frac{1}{2 \left(e^{x} + 1\right)} - \frac{1}{2 \left(e^{x} - 1\right)}$$
1 1
- ---------- - -----------
/ x\ / x\
2*\1 + e / 2*\-1 + e /
General simplification
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-1 --------- 2*sinh(x)
$$- \frac{1}{2 \sinh{\left(x \right)}}$$
-1/(2*sinh(x))
Combining rational expressions
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x
-e
------------------
/ x\ / x\
\1 + e /*\-1 + e /
$$- \frac{e^{x}}{\left(e^{x} - 1\right) \left(e^{x} + 1\right)}$$
-exp(x)/((1 + exp(x))*(-1 + exp(x)))
Combinatorics
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x
-e
------------------
/ x\ / x\
\1 + e /*\-1 + e /
$$- \frac{e^{x}}{\left(e^{x} - 1\right) \left(e^{x} + 1\right)}$$
-exp(x)/((1 + exp(x))*(-1 + exp(x)))
Assemble expression
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x x
e e
-------- - ---------
x x
2 + 2*e -2 + 2*e
$$\frac{e^{x}}{2 e^{x} + 2} - \frac{e^{x}}{2 e^{x} - 2}$$
exp(x)/(2 + 2*exp(x)) - exp(x)/(-2 + 2*exp(x))
Rational denominator
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/ x\ x / x\ x
\-2 + 2*e /*e - \2 + 2*e /*e
------------------------------
/ x\ / x\
\-2 + 2*e /*\2 + 2*e /
$$\frac{\left(2 e^{x} - 2\right) e^{x} - \left(2 e^{x} + 2\right) e^{x}}{\left(2 e^{x} - 2\right) \left(2 e^{x} + 2\right)}$$
((-2 + 2*exp(x))*exp(x) - (2 + 2*exp(x))*exp(x))/((-2 + 2*exp(x))*(2 + 2*exp(x)))
Powers
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x x
e e
-------- - ---------
x x
2 + 2*e -2 + 2*e
$$\frac{e^{x}}{2 e^{x} + 2} - \frac{e^{x}}{2 e^{x} - 2}$$
exp(x)/(2 + 2*exp(x)) - exp(x)/(-2 + 2*exp(x))
Numerical answer
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exp(x)/(2.0 + 2.0*2.71828182845905^x) - exp(x)/(-2.0 + 2.0*2.71828182845905^x)
exp(x)/(2.0 + 2.0*2.71828182845905^x) - exp(x)/(-2.0 + 2.0*2.71828182845905^x)
Common denominator
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x
-e
---------
2*x
-1 + e
$$- \frac{e^{x}}{e^{2 x} - 1}$$
-exp(x)/(-1 + exp(2*x))
Trigonometric part
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x x
e e
-------- - ---------
x x
2 + 2*e -2 + 2*e
$$\frac{e^{x}}{2 e^{x} + 2} - \frac{e^{x}}{2 e^{x} - 2}$$
-(cosh(x) + sinh(x))
---------------------------
2*x
-1 + (cosh(1) + sinh(1))
$$- \frac{\sinh{\left(x \right)} + \cosh{\left(x \right)}}{\left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{2 x} - 1}$$
cosh(x) + sinh(x) cosh(x) + sinh(x) ------------------------- - -------------------------- 2 + 2*cosh(x) + 2*sinh(x) -2 + 2*cosh(x) + 2*sinh(x)
$$\frac{\sinh{\left(x \right)} + \cosh{\left(x \right)}}{2 \sinh{\left(x \right)} + 2 \cosh{\left(x \right)} + 2} - \frac{\sinh{\left(x \right)} + \cosh{\left(x \right)}}{2 \sinh{\left(x \right)} + 2 \cosh{\left(x \right)} - 2}$$
cosh(x) + sinh(x) cosh(x) + sinh(x)
-------------------------- - ---------------------------
x x
2 + 2*(cosh(1) + sinh(1)) -2 + 2*(cosh(1) + sinh(1))
$$\frac{\sinh{\left(x \right)} + \cosh{\left(x \right)}}{2 \left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{x} + 2} - \frac{\sinh{\left(x \right)} + \cosh{\left(x \right)}}{2 \left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{x} - 2}$$
-(cosh(x) + sinh(x)) -------------------------- -1 + cosh(2*x) + sinh(2*x)
$$- \frac{\sinh{\left(x \right)} + \cosh{\left(x \right)}}{\sinh{\left(2 x \right)} + \cosh{\left(2 x \right)} - 1}$$
-(cosh(x) + sinh(x))/(-1 + cosh(2*x) + sinh(2*x))