Mister Exam
Expression simplification/
Least common denominator/
sqrt(4/x+1)*x+2*log(sqrt(4/x+1)+1)-2*log(sqrt(4/x+1)-1)
Least common denominator sqrt(4/x+1)*x+2*log(sqrt(4/x+1)+1)-2*log(sqrt(4/x+1)-1)
The solution
You have entered
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_______ / _______ \ / _______ \ / 4 | / 4 | | / 4 | / - + 1 *x + 2*log| / - + 1 + 1| - 2*log| / - + 1 - 1| \/ x \\/ x / \\/ x /
$$\left(x \sqrt{1 + \frac{4}{x}} + 2 \log{\left(\sqrt{1 + \frac{4}{x}} + 1 \right)}\right) - 2 \log{\left(\sqrt{1 + \frac{4}{x}} - 1 \right)}$$
sqrt(4/x + 1)*x + 2*log(sqrt(4/x + 1) + 1) - 2*log(sqrt(4/x + 1) - 1)
General simplification
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/ _______\ / _______\ _______
| / 4 + x | | / 4 + x | / 4 + x
- 2*log|-1 + / ----- | + 2*log|1 + / ----- | + x* / -----
\ \/ x / \ \/ x / \/ x
$$x \sqrt{\frac{x + 4}{x}} - 2 \log{\left(\sqrt{\frac{x + 4}{x}} - 1 \right)} + 2 \log{\left(\sqrt{\frac{x + 4}{x}} + 1 \right)}$$
-2*log(-1 + sqrt((4 + x)/x)) + 2*log(1 + sqrt((4 + x)/x)) + x*sqrt((4 + x)/x)
Common denominator
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/ _______\ / _______\ _______
| / 4 | | / 4 | / 4
- 2*log|-1 + / 1 + - | + 2*log|1 + / 1 + - | + x* / 1 + -
\ \/ x / \ \/ x / \/ x
$$x \sqrt{1 + \frac{4}{x}} - 2 \log{\left(\sqrt{1 + \frac{4}{x}} - 1 \right)} + 2 \log{\left(\sqrt{1 + \frac{4}{x}} + 1 \right)}$$
-2*log(-1 + sqrt(1 + 4/x)) + 2*log(1 + sqrt(1 + 4/x)) + x*sqrt(1 + 4/x)
Rational denominator
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/ _______\ / _______\ _______
| / 4 | | / 4 | / 4
- 2*log|-1 + / 1 + - | + 2*log|1 + / 1 + - | + x* / 1 + -
\ \/ x / \ \/ x / \/ x
$$x \sqrt{1 + \frac{4}{x}} - 2 \log{\left(\sqrt{1 + \frac{4}{x}} - 1 \right)} + 2 \log{\left(\sqrt{1 + \frac{4}{x}} + 1 \right)}$$
-2*log(-1 + sqrt(1 + 4/x)) + 2*log(1 + sqrt(1 + 4/x)) + x*sqrt(1 + 4/x)
Trigonometric part
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/ _______\ / _______\ _______
| / 4 | | / 4 | / 4
- 2*log|-1 + / 1 + - | + 2*log|1 + / 1 + - | + x* / 1 + -
\ \/ x / \ \/ x / \/ x
$$x \sqrt{1 + \frac{4}{x}} - 2 \log{\left(\sqrt{1 + \frac{4}{x}} - 1 \right)} + 2 \log{\left(\sqrt{1 + \frac{4}{x}} + 1 \right)}$$
-2*log(-1 + sqrt(1 + 4/x)) + 2*log(1 + sqrt(1 + 4/x)) + x*sqrt(1 + 4/x)
Combining rational expressions
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/ _______\ / _______\ _______
| / 4 + x | | / 4 + x | / 4 + x
- 2*log|-1 + / ----- | + 2*log|1 + / ----- | + x* / -----
\ \/ x / \ \/ x / \/ x
$$x \sqrt{\frac{x + 4}{x}} - 2 \log{\left(\sqrt{\frac{x + 4}{x}} - 1 \right)} + 2 \log{\left(\sqrt{\frac{x + 4}{x}} + 1 \right)}$$
-2*log(-1 + sqrt((4 + x)/x)) + 2*log(1 + sqrt((4 + x)/x)) + x*sqrt((4 + x)/x)
Combinatorics
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/ _______\ / _______\ _______
| / 4 | | / 4 | / 4
- 2*log|-1 + / 1 + - | + 2*log|1 + / 1 + - | + x* / 1 + -
\ \/ x / \ \/ x / \/ x
$$x \sqrt{1 + \frac{4}{x}} - 2 \log{\left(\sqrt{1 + \frac{4}{x}} - 1 \right)} + 2 \log{\left(\sqrt{1 + \frac{4}{x}} + 1 \right)}$$
-2*log(-1 + sqrt(1 + 4/x)) + 2*log(1 + sqrt(1 + 4/x)) + x*sqrt(1 + 4/x)
Numerical answer
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2.0*log(sqrt(4/x + 1) + 1) - 2.0*log(sqrt(4/x + 1) - 1) + 2.0*x*(0.25 + 1/x)^0.5
2.0*log(sqrt(4/x + 1) + 1) - 2.0*log(sqrt(4/x + 1) - 1) + 2.0*x*(0.25 + 1/x)^0.5
Powers
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/ _______\ / _______\ _______
| / 4 | | / 4 | / 4
- 2*log|-1 + / 1 + - | + 2*log|1 + / 1 + - | + x* / 1 + -
\ \/ x / \ \/ x / \/ x
$$x \sqrt{1 + \frac{4}{x}} - 2 \log{\left(\sqrt{1 + \frac{4}{x}} - 1 \right)} + 2 \log{\left(\sqrt{1 + \frac{4}{x}} + 1 \right)}$$
-2*log(-1 + sqrt(1 + 4/x)) + 2*log(1 + sqrt(1 + 4/x)) + x*sqrt(1 + 4/x)
Assemble expression
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/ _______ \ / _______ \ _______
| / 4 | | / 4 | / 4
- 2*log| / - + 1 - 1| + 2*log| / - + 1 + 1| + x* / 1 + -
\\/ x / \\/ x / \/ x
$$x \sqrt{1 + \frac{4}{x}} - 2 \log{\left(\sqrt{1 + \frac{4}{x}} - 1 \right)} + 2 \log{\left(\sqrt{1 + \frac{4}{x}} + 1 \right)}$$
-2*log(sqrt(4/x + 1) - 1) + 2*log(sqrt(4/x + 1) + 1) + x*sqrt(1 + 4/x)