Mister Exam
Expression simplification/
Least common denominator/
tan(x^2)/(x*log(2))-x/(sqrt(x^2)*sqrt(1-x^2))+2*x*(1+tan(x^2)^2)*log(x)/log(2)
Least common denominator tan(x^2)/(x*log(2))-x/(sqrt(x^2)*sqrt(1-x^2))+2*x*(1+tan(x^2)^2)*log(x)/log(2)
The solution
You have entered
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/ 2\ / 2/ 2\\
tan\x / x 2*x*\1 + tan \x //*log(x)
-------- - ------------------- + -------------------------
x*log(2) ____ ________ log(2)
/ 2 / 2
\/ x *\/ 1 - x
$$\frac{2 x \left(\tan^{2}{\left(x^{2} \right)} + 1\right) \log{\left(x \right)}}{\log{\left(2 \right)}} + \left(- \frac{x}{\sqrt{1 - x^{2}} \sqrt{x^{2}}} + \frac{\tan{\left(x^{2} \right)}}{x \log{\left(2 \right)}}\right)$$
tan(x^2)/((x*log(2))) - x/(sqrt(x^2)*sqrt(1 - x^2)) + (((2*x)*(1 + tan(x^2)^2))*log(x))/log(2)
General simplification
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/ 2\
tan\x / x 2*x*log(x)
-------- - ------------------- + ---------------
x*log(2) ____ ________ 2/ 2\
/ 2 / 2 cos \x /*log(2)
\/ x *\/ 1 - x
$$\frac{2 x \log{\left(x \right)}}{\log{\left(2 \right)} \cos^{2}{\left(x^{2} \right)}} - \frac{x}{\sqrt{1 - x^{2}} \sqrt{x^{2}}} + \frac{\tan{\left(x^{2} \right)}}{x \log{\left(2 \right)}}$$
tan(x^2)/(x*log(2)) - x/(sqrt(x^2)*sqrt(1 - x^2)) + 2*x*log(x)/(cos(x^2)^2*log(2))
Numerical answer
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1.44269504088896*tan(x^2)/x - x*(x^2)^(-0.5)*(1.0 - x^2)^(-0.5) + 2.88539008177793*x*(1.0 + tan(x^2)^2)*log(x)
1.44269504088896*tan(x^2)/x - x*(x^2)^(-0.5)*(1.0 - x^2)^(-0.5) + 2.88539008177793*x*(1.0 + tan(x^2)^2)*log(x)
Combinatorics
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/ ____ ________ ____ ________ ____ ________ \
| 2 / 2 / 2 / 2\ 2 / 2 / 2 2 / 2 / 2 2/ 2\ |
-\x *log(2) - \/ x *\/ 1 - x *tan\x / - 2*x *\/ x *\/ 1 - x *log(x) - 2*x *\/ x *\/ 1 - x *tan \x /*log(x)/
------------------------------------------------------------------------------------------------------------------------
____
/ 2 ___________________
x*\/ x *\/ -(1 + x)*(-1 + x) *log(2)
$$- \frac{- 2 x^{2} \sqrt{1 - x^{2}} \sqrt{x^{2}} \log{\left(x \right)} \tan^{2}{\left(x^{2} \right)} - 2 x^{2} \sqrt{1 - x^{2}} \sqrt{x^{2}} \log{\left(x \right)} + x^{2} \log{\left(2 \right)} - \sqrt{1 - x^{2}} \sqrt{x^{2}} \tan{\left(x^{2} \right)}}{x \sqrt{- \left(x - 1\right) \left(x + 1\right)} \sqrt{x^{2}} \log{\left(2 \right)}}$$
-(x^2*log(2) - sqrt(x^2)*sqrt(1 - x^2)*tan(x^2) - 2*x^2*sqrt(x^2)*sqrt(1 - x^2)*log(x) - 2*x^2*sqrt(x^2)*sqrt(1 - x^2)*tan(x^2)^2*log(x))/(x*sqrt(x^2)*sqrt(-(1 + x)*(-1 + x))*log(2))
Rational denominator
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____ ________
4 / 2\ 2 / 2\ 4 6 2 / 2 / 2 2 4 2/ 2\ 6 2/ 2\
x *log(2)*tan\x / - x *log(2)*tan\x / - 2*x *log(2)*log(x) + 2*x *log(2)*log(x) + x *\/ x *\/ 1 - x *log (2) - 2*x *tan \x /*log(2)*log(x) + 2*x *tan \x /*log(2)*log(x)
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------
3 / 2\ 2
x *\-1 + x /*log (2)
$$\frac{2 x^{6} \log{\left(2 \right)} \log{\left(x \right)} \tan^{2}{\left(x^{2} \right)} + 2 x^{6} \log{\left(2 \right)} \log{\left(x \right)} - 2 x^{4} \log{\left(2 \right)} \log{\left(x \right)} \tan^{2}{\left(x^{2} \right)} - 2 x^{4} \log{\left(2 \right)} \log{\left(x \right)} + x^{4} \log{\left(2 \right)} \tan{\left(x^{2} \right)} + x^{2} \sqrt{1 - x^{2}} \sqrt{x^{2}} \log{\left(2 \right)}^{2} - x^{2} \log{\left(2 \right)} \tan{\left(x^{2} \right)}}{x^{3} \left(x^{2} - 1\right) \log{\left(2 \right)}^{2}}$$
(x^4*log(2)*tan(x^2) - x^2*log(2)*tan(x^2) - 2*x^4*log(2)*log(x) + 2*x^6*log(2)*log(x) + x^2*sqrt(x^2)*sqrt(1 - x^2)*log(2)^2 - 2*x^4*tan(x^2)^2*log(2)*log(x) + 2*x^6*tan(x^2)^2*log(2)*log(x))/(x^3*(-1 + x^2)*log(2)^2)
Combining rational expressions
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____ ________ ____ ________
2 / 2 / 2 / 2\ 2 / 2 / 2 / 2/ 2\\
- x *log(2) + \/ x *\/ 1 - x *tan\x / + 2*x *\/ x *\/ 1 - x *\1 + tan \x //*log(x)
------------------------------------------------------------------------------------------
____ ________
/ 2 / 2
x*\/ x *\/ 1 - x *log(2)
$$\frac{2 x^{2} \sqrt{1 - x^{2}} \left(\tan^{2}{\left(x^{2} \right)} + 1\right) \sqrt{x^{2}} \log{\left(x \right)} - x^{2} \log{\left(2 \right)} + \sqrt{1 - x^{2}} \sqrt{x^{2}} \tan{\left(x^{2} \right)}}{x \sqrt{1 - x^{2}} \sqrt{x^{2}} \log{\left(2 \right)}}$$
(-x^2*log(2) + sqrt(x^2)*sqrt(1 - x^2)*tan(x^2) + 2*x^2*sqrt(x^2)*sqrt(1 - x^2)*(1 + tan(x^2)^2)*log(x))/(x*sqrt(x^2)*sqrt(1 - x^2)*log(2))
Common denominator
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/ ____ ________ ____ ________ ____ ________ \
| 2 / 2 / 2 / 2\ 2 / 2 / 2 2 / 2 / 2 2/ 2\ |
-\x *log(2) - \/ x *\/ 1 - x *tan\x / - 2*x *\/ x *\/ 1 - x *log(x) - 2*x *\/ x *\/ 1 - x *tan \x /*log(x)/
------------------------------------------------------------------------------------------------------------------------
____ ________
/ 2 / 2
x*\/ x *\/ 1 - x *log(2)
$$- \frac{- 2 x^{2} \sqrt{1 - x^{2}} \sqrt{x^{2}} \log{\left(x \right)} \tan^{2}{\left(x^{2} \right)} - 2 x^{2} \sqrt{1 - x^{2}} \sqrt{x^{2}} \log{\left(x \right)} + x^{2} \log{\left(2 \right)} - \sqrt{1 - x^{2}} \sqrt{x^{2}} \tan{\left(x^{2} \right)}}{x \sqrt{1 - x^{2}} \sqrt{x^{2}} \log{\left(2 \right)}}$$
-(x^2*log(2) - sqrt(x^2)*sqrt(1 - x^2)*tan(x^2) - 2*x^2*sqrt(x^2)*sqrt(1 - x^2)*log(x) - 2*x^2*sqrt(x^2)*sqrt(1 - x^2)*tan(x^2)^2*log(x))/(x*sqrt(x^2)*sqrt(1 - x^2)*log(2))
Powers
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/ 2\ / 2/ 2\\
x tan\x / 2*x*\1 + tan \x //*log(x)
- ---------------- + -------- + -------------------------
_____________ x*log(2) log(2)
/ 2 / 2\
\/ x *\1 - x /
$$\frac{2 x \left(\tan^{2}{\left(x^{2} \right)} + 1\right) \log{\left(x \right)}}{\log{\left(2 \right)}} - \frac{x}{\sqrt{x^{2} \left(1 - x^{2}\right)}} + \frac{\tan{\left(x^{2} \right)}}{x \log{\left(2 \right)}}$$
/ 2\ / 2/ 2\\
tan\x / x 2*x*\1 + tan \x //*log(x)
-------- - ------------------- + -------------------------
x*log(2) ____ ________ log(2)
/ 2 / 2
\/ x *\/ 1 - x
$$\frac{2 x \left(\tan^{2}{\left(x^{2} \right)} + 1\right) \log{\left(x \right)}}{\log{\left(2 \right)}} - \frac{x}{\sqrt{1 - x^{2}} \sqrt{x^{2}}} + \frac{\tan{\left(x^{2} \right)}}{x \log{\left(2 \right)}}$$
/ 2\
| / 2 2\ |
| | I*x -I*x | |
| \- e + e / |
2*x*|1 - -------------------|*log(x)
| 2 |
| / 2 2\ | / 2 2\
| | I*x -I*x | | | I*x -I*x |
x \ \e + e / / I*\- e + e /
- ------------------- + ------------------------------------ + -------------------------
____ ________ log(2) / 2 2\
/ 2 / 2 | I*x -I*x |
\/ x *\/ 1 - x x*\e + e /*log(2)
$$\frac{2 x \left(- \frac{\left(- e^{i x^{2}} + e^{- i x^{2}}\right)^{2}}{\left(e^{i x^{2}} + e^{- i x^{2}}\right)^{2}} + 1\right) \log{\left(x \right)}}{\log{\left(2 \right)}} - \frac{x}{\sqrt{1 - x^{2}} \sqrt{x^{2}}} + \frac{i \left(- e^{i x^{2}} + e^{- i x^{2}}\right)}{x \left(e^{i x^{2}} + e^{- i x^{2}}\right) \log{\left(2 \right)}}$$
-x/(sqrt(x^2)*sqrt(1 - x^2)) + 2*x*(1 - (-exp(i*x^2) + exp(-i*x^2))^2/(exp(i*x^2) + exp(-i*x^2))^2)*log(x)/log(2) + i*(-exp(i*x^2) + exp(-i*x^2))/(x*(exp(i*x^2) + exp(-i*x^2))*log(2))
Assemble expression
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/ 2\ / 2/ 2\\
tan\x / x 2*x*\1 + tan \x //*log(x)
-------- - ------------------- + -------------------------
x*log(2) ____ ________ log(2)
/ 2 / 2
\/ x *\/ 1 - x
$$\frac{2 x \left(\tan^{2}{\left(x^{2} \right)} + 1\right) \log{\left(x \right)}}{\log{\left(2 \right)}} - \frac{x}{\sqrt{1 - x^{2}} \sqrt{x^{2}}} + \frac{\tan{\left(x^{2} \right)}}{x \log{\left(2 \right)}}$$
tan(x^2)/(x*log(2)) - x/(sqrt(x^2)*sqrt(1 - x^2)) + 2*x*(1 + tan(x^2)^2)*log(x)/log(2)
Trigonometric part
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/ 2/ 2\\
| sin \x /|
2*x*|1 + --------|*log(x)
/ 2\ | 2/ 2\|
x sin\x / \ cos \x //
- ------------------- + ---------------- + -------------------------
____ ________ / 2\ log(2)
/ 2 / 2 x*cos\x /*log(2)
\/ x *\/ 1 - x
$$\frac{2 x \left(\frac{\sin^{2}{\left(x^{2} \right)}}{\cos^{2}{\left(x^{2} \right)}} + 1\right) \log{\left(x \right)}}{\log{\left(2 \right)}} - \frac{x}{\sqrt{1 - x^{2}} \sqrt{x^{2}}} + \frac{\sin{\left(x^{2} \right)}}{x \log{\left(2 \right)} \cos{\left(x^{2} \right)}}$$
/ 2\ 2/ 2\
x sec\x / 2*x*sec \x /*log(x)
- ------------------- + --------------------- + -------------------
____ ________ / 2 pi\ log(2)
/ 2 / 2 x*log(2)*sec|x - --|
\/ x *\/ 1 - x \ 2 /
$$\frac{2 x \log{\left(x \right)} \sec^{2}{\left(x^{2} \right)}}{\log{\left(2 \right)}} - \frac{x}{\sqrt{1 - x^{2}} \sqrt{x^{2}}} + \frac{\sec{\left(x^{2} \right)}}{x \log{\left(2 \right)} \sec{\left(x^{2} - \frac{\pi}{2} \right)}}$$
/ 2\
tan\x / x 2*x*log(x)
-------- - ------------------- + ---------------
x*log(2) ____ ________ 2/ 2\
/ 2 / 2 cos \x /*log(2)
\/ x *\/ 1 - x
$$\frac{2 x \log{\left(x \right)}}{\log{\left(2 \right)} \cos^{2}{\left(x^{2} \right)}} - \frac{x}{\sqrt{1 - x^{2}} \sqrt{x^{2}}} + \frac{\tan{\left(x^{2} \right)}}{x \log{\left(2 \right)}}$$
/ 2\ 2/ 2\
tan\x / x 2*x*sec \x /*log(x)
-------- - ------------------- + -------------------
x*log(2) ____ ________ log(2)
/ 2 / 2
\/ x *\/ 1 - x
$$\frac{2 x \log{\left(x \right)} \sec^{2}{\left(x^{2} \right)}}{\log{\left(2 \right)}} - \frac{x}{\sqrt{1 - x^{2}} \sqrt{x^{2}}} + \frac{\tan{\left(x^{2} \right)}}{x \log{\left(2 \right)}}$$
/ 2 pi\
cos|x - --|
x \ 2 / 2*x*log(x)
- ------------------- + ---------------- + ---------------
____ ________ / 2\ 2/ 2\
/ 2 / 2 x*cos\x /*log(2) cos \x /*log(2)
\/ x *\/ 1 - x
$$\frac{2 x \log{\left(x \right)}}{\log{\left(2 \right)} \cos^{2}{\left(x^{2} \right)}} - \frac{x}{\sqrt{1 - x^{2}} \sqrt{x^{2}}} + \frac{\cos{\left(x^{2} - \frac{\pi}{2} \right)}}{x \log{\left(2 \right)} \cos{\left(x^{2} \right)}}$$
/ 2/ 2\\
| sec \x /|
2*x*|1 + --------|*log(x)
/ 2\ | 2/ 2\|
x sec\x / \ csc \x //
- ------------------- + ---------------- + -------------------------
____ ________ / 2\ log(2)
/ 2 / 2 x*csc\x /*log(2)
\/ x *\/ 1 - x
$$\frac{2 x \left(1 + \frac{\sec^{2}{\left(x^{2} \right)}}{\csc^{2}{\left(x^{2} \right)}}\right) \log{\left(x \right)}}{\log{\left(2 \right)}} - \frac{x}{\sqrt{1 - x^{2}} \sqrt{x^{2}}} + \frac{\sec{\left(x^{2} \right)}}{x \log{\left(2 \right)} \csc{\left(x^{2} \right)}}$$
2
/ / 2\\
| 2|x ||
2*x*|1 + cot |--|| *log(x)
1 x \ \2 //
---------------- - ------------------- + --------------------------
/ 2\ ____ ________ 2
x*cot\x /*log(2) / 2 / 2 / / 2\\
\/ x *\/ 1 - x | 2|x ||
|-1 + cot |--|| *log(2)
\ \2 //
$$\frac{2 x \left(\cot^{2}{\left(\frac{x^{2}}{2} \right)} + 1\right)^{2} \log{\left(x \right)}}{\left(\cot^{2}{\left(\frac{x^{2}}{2} \right)} - 1\right)^{2} \log{\left(2 \right)}} - \frac{x}{\sqrt{1 - x^{2}} \sqrt{x^{2}}} + \frac{1}{x \log{\left(2 \right)} \cot{\left(x^{2} \right)}}$$
/ 4/ 2\\
| 4*sin \x /|
2*x*|1 + ----------|*log(x)
| 2/ 2\| 2/ 2\
x \ sin \2*x // 2*sin \x /
- ------------------- + --------------------------- + ------------------
____ ________ log(2) / 2\
/ 2 / 2 x*log(2)*sin\2*x /
\/ x *\/ 1 - x
$$\frac{2 x \left(\frac{4 \sin^{4}{\left(x^{2} \right)}}{\sin^{2}{\left(2 x^{2} \right)}} + 1\right) \log{\left(x \right)}}{\log{\left(2 \right)}} - \frac{x}{\sqrt{1 - x^{2}} \sqrt{x^{2}}} + \frac{2 \sin^{2}{\left(x^{2} \right)}}{x \log{\left(2 \right)} \sin{\left(2 x^{2} \right)}}$$
/ 1 \
2*x*|1 + --------|*log(x)
| 2/ 2\|
1 x \ cot \x //
---------------- - ------------------- + -------------------------
/ 2\ ____ ________ log(2)
x*cot\x /*log(2) / 2 / 2
\/ x *\/ 1 - x
$$\frac{2 x \left(1 + \frac{1}{\cot^{2}{\left(x^{2} \right)}}\right) \log{\left(x \right)}}{\log{\left(2 \right)}} - \frac{x}{\sqrt{1 - x^{2}} \sqrt{x^{2}}} + \frac{1}{x \log{\left(2 \right)} \cot{\left(x^{2} \right)}}$$
/ 2\ / 2/ 2\\
tan\x / x 2*x*\1 + tan \x //*log(x)
-------- - ------------------- + -------------------------
x*log(2) ____ ________ log(2)
/ 2 / 2
\/ x *\/ 1 - x
$$\frac{2 x \left(\tan^{2}{\left(x^{2} \right)} + 1\right) \log{\left(x \right)}}{\log{\left(2 \right)}} - \frac{x}{\sqrt{1 - x^{2}} \sqrt{x^{2}}} + \frac{\tan{\left(x^{2} \right)}}{x \log{\left(2 \right)}}$$
2
/ / 2\\
| 2|x ||
/ 2\ 2*x*|1 + tan |--|| *log(x)
tan\x / x \ \2 //
-------- - ------------------- + --------------------------
x*log(2) ____ ________ 2
/ 2 / 2 / / 2\\
\/ x *\/ 1 - x | 2|x ||
|1 - tan |--|| *log(2)
\ \2 //
$$\frac{2 x \left(\tan^{2}{\left(\frac{x^{2}}{2} \right)} + 1\right)^{2} \log{\left(x \right)}}{\left(1 - \tan^{2}{\left(\frac{x^{2}}{2} \right)}\right)^{2} \log{\left(2 \right)}} - \frac{x}{\sqrt{1 - x^{2}} \sqrt{x^{2}}} + \frac{\tan{\left(x^{2} \right)}}{x \log{\left(2 \right)}}$$
/ 2/ 2\ \
| sec \x / |
2*x*|1 + -------------|*log(x)
| 2/ 2 pi\|
/ 2\ | sec |x - --||
x sec\x / \ \ 2 //
- ------------------- + --------------------- + ------------------------------
____ ________ / 2 pi\ log(2)
/ 2 / 2 x*log(2)*sec|x - --|
\/ x *\/ 1 - x \ 2 /
$$\frac{2 x \left(\frac{\sec^{2}{\left(x^{2} \right)}}{\sec^{2}{\left(x^{2} - \frac{\pi}{2} \right)}} + 1\right) \log{\left(x \right)}}{\log{\left(2 \right)}} - \frac{x}{\sqrt{1 - x^{2}} \sqrt{x^{2}}} + \frac{\sec{\left(x^{2} \right)}}{x \log{\left(2 \right)} \sec{\left(x^{2} - \frac{\pi}{2} \right)}}$$
2/ 2\
x 2*x*log(x) 2*sin \x /
- ------------------- + -------------------- + ------------------
____ ________ 2/ 2 pi\ / 2\
/ 2 / 2 log(2)*sin |x + --| x*log(2)*sin\2*x /
\/ x *\/ 1 - x \ 2 /
$$\frac{2 x \log{\left(x \right)}}{\log{\left(2 \right)} \sin^{2}{\left(x^{2} + \frac{\pi}{2} \right)}} - \frac{x}{\sqrt{1 - x^{2}} \sqrt{x^{2}}} + \frac{2 \sin^{2}{\left(x^{2} \right)}}{x \log{\left(2 \right)} \sin{\left(2 x^{2} \right)}}$$
/ 2/ 2 pi\\
| cos |x - --||
| \ 2 /|
/ 2 pi\ 2*x*|1 + -------------|*log(x)
cos|x - --| | 2/ 2\ |
x \ 2 / \ cos \x / /
- ------------------- + ---------------- + ------------------------------
____ ________ / 2\ log(2)
/ 2 / 2 x*cos\x /*log(2)
\/ x *\/ 1 - x
$$\frac{2 x \left(1 + \frac{\cos^{2}{\left(x^{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x^{2} \right)}}\right) \log{\left(x \right)}}{\log{\left(2 \right)}} - \frac{x}{\sqrt{1 - x^{2}} \sqrt{x^{2}}} + \frac{\cos{\left(x^{2} - \frac{\pi}{2} \right)}}{x \log{\left(2 \right)} \cos{\left(x^{2} \right)}}$$
/ 2/pi 2\\
| csc |-- - x ||
| \2 /|
/pi 2\ 2*x*|1 + -------------|*log(x)
csc|-- - x | | 2/ 2\ |
x \2 / \ csc \x / /
- ------------------- + ---------------- + ------------------------------
____ ________ / 2\ log(2)
/ 2 / 2 x*csc\x /*log(2)
\/ x *\/ 1 - x
$$\frac{2 x \left(1 + \frac{\csc^{2}{\left(- x^{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(x^{2} \right)}}\right) \log{\left(x \right)}}{\log{\left(2 \right)}} - \frac{x}{\sqrt{1 - x^{2}} \sqrt{x^{2}}} + \frac{\csc{\left(- x^{2} + \frac{\pi}{2} \right)}}{x \log{\left(2 \right)} \csc{\left(x^{2} \right)}}$$
/pi 2\ 2/pi 2\
csc|-- - x | 2*x*csc |-- - x |*log(x)
x \2 / \2 /
- ------------------- + ---------------- + ------------------------
____ ________ / 2\ log(2)
/ 2 / 2 x*csc\x /*log(2)
\/ x *\/ 1 - x
$$\frac{2 x \log{\left(x \right)} \csc^{2}{\left(- x^{2} + \frac{\pi}{2} \right)}}{\log{\left(2 \right)}} - \frac{x}{\sqrt{1 - x^{2}} \sqrt{x^{2}}} + \frac{\csc{\left(- x^{2} + \frac{\pi}{2} \right)}}{x \log{\left(2 \right)} \csc{\left(x^{2} \right)}}$$
-x/(sqrt(x^2)*sqrt(1 - x^2)) + csc(pi/2 - x^2)/(x*csc(x^2)*log(2)) + 2*x*csc(pi/2 - x^2)^2*log(x)/log(2)