Mister Exam
Expression simplification/
Fraction Decomposition into the simple/
(4x(x+sqrt(x^2-1)^2))/((x+(sqrtx^2-1))^4-1)
How do you (4x(x+sqrt(x^2-1)^2))/((x+(sqrtx^2-1))^4-1) in partial fractions?
The solution
You have entered
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/ 2\
| ________ |
| / 2 |
4*x*\x + \/ x - 1 /
----------------------
4
/ 2 \
| ___ |
\x + \/ x - 1/ - 1
$$\frac{4 x \left(x + \left(\sqrt{x^{2} - 1}\right)^{2}\right)}{\left(x + \left(\left(\sqrt{x}\right)^{2} - 1\right)\right)^{4} - 1}$$
((4*x)*(x + (sqrt(x^2 - 1))^2))/((x + (sqrt(x))^2 - 1)^4 - 1)
Fraction decomposition
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1/(2*(-1 + x)) - (-2 + x)/(2*(1 - 2*x + 2*x^2))
$$- \frac{x - 2}{2 \left(2 x^{2} - 2 x + 1\right)} + \frac{1}{2 \left(x - 1\right)}$$
1 -2 + x
---------- - ------------------
2*(-1 + x) / 2\
2*\1 - 2*x + 2*x /
General simplification
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/ 2\
4*x*\-1 + x + x /
-----------------
4
-1 + (-1 + 2*x)
$$\frac{4 x \left(x^{2} + x - 1\right)}{\left(2 x - 1\right)^{4} - 1}$$
4*x*(-1 + x + x^2)/(-1 + (-1 + 2*x)^4)
Numerical answer
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4.0*x*(x + (-1.0 + x^2)^1.0)/(-1.0 + (-1.0 + x + x^1.0)^4)
4.0*x*(x + (-1.0 + x^2)^1.0)/(-1.0 + (-1.0 + x + x^1.0)^4)
Trigonometric part
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/ 2\
4*x*\-1 + x + x /
-----------------
4
-1 + (-1 + 2*x)
$$\frac{4 x \left(x^{2} + x - 1\right)}{\left(2 x - 1\right)^{4} - 1}$$
4*x*(-1 + x + x^2)/(-1 + (-1 + 2*x)^4)
Combining rational expressions
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/ 2\
4*x*\-1 + x + x /
-----------------
4
-1 + (-1 + 2*x)
$$\frac{4 x \left(x^{2} + x - 1\right)}{\left(2 x - 1\right)^{4} - 1}$$
4*x*(-1 + x + x^2)/(-1 + (-1 + 2*x)^4)
Assemble expression
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/ 2\
4*x*\-1 + x + x /
-----------------
4
-1 + (-1 + 2*x)
$$\frac{4 x \left(x^{2} + x - 1\right)}{\left(2 x - 1\right)^{4} - 1}$$
4*x*(-1 + x + x^2)/(-1 + (-1 + 2*x)^4)
Rational denominator
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2
-1 + x + x
----------------------
2 3
-2 - 8*x + 4*x + 6*x
$$\frac{x^{2} + x - 1}{4 x^{3} - 8 x^{2} + 6 x - 2}$$
(-1 + x + x^2)/(-2 - 8*x^2 + 4*x^3 + 6*x)
Powers
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/ 2\
4*x*\-1 + x + x /
-----------------
4
-1 + (-1 + 2*x)
$$\frac{4 x \left(x^{2} + x - 1\right)}{\left(2 x - 1\right)^{4} - 1}$$
4*x*(-1 + x + x^2)/(-1 + (-1 + 2*x)^4)
Common denominator
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2
-1 + x + x
----------------------
2 3
-2 - 8*x + 4*x + 6*x
$$\frac{x^{2} + x - 1}{4 x^{3} - 8 x^{2} + 6 x - 2}$$
(-1 + x + x^2)/(-2 - 8*x^2 + 4*x^3 + 6*x)
Combinatorics
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2
-1 + x + x
---------------------------
/ 2\
2*(-1 + x)*\1 - 2*x + 2*x /
$$\frac{x^{2} + x - 1}{2 \left(x - 1\right) \left(2 x^{2} - 2 x + 1\right)}$$
(-1 + x + x^2)/(2*(-1 + x)*(1 - 2*x + 2*x^2))