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- Improper integral
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How to use it?
- Integral of d{x}:
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Integral of 1/sqrt(x^2+1)
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Integral of x^3*ln(x)
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Integral of x^2*lnx
- Integral of 1/(x(1+x^2))
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Identical expressions
- (x- six)/(x^ four +6x+ eight)
- (x minus 6) divide by (x to the power of 4 plus 6x plus 8)
- (x minus six) divide by (x to the power of four plus 6x plus eight)
- (x-6)/(x4+6x+8)
- x-6/x4+6x+8
- (x-6)/(x⁴+6x+8)
- x-6/x^4+6x+8
- (x-6) divide by (x^4+6x+8)
- (x-6)/(x^4+6x+8)dx
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Similar expressions
- (x-6)/(x^4-6x+8)
- (x-6)/(x^4+6x-8)
- (x+6)/(x^4+6x+8)
Integral of (x-6)/(x^4+6x+8) dx
The solution
You have entered
[src]
1 / | | x - 6 | ------------ dx | 4 | x + 6*x + 8 | / 0
$$\int\limits_{0}^{1} \frac{x - 6}{\left(x^{4} + 6 x\right) + 8}\, dx$$
Integral((x - 6)/(x^4 + 6*x + 8), (x, 0, 1))
Detail solution
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Rewrite the integrand:
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Integrate term-by-term:
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Don't know the steps in finding this integral.
But the integral is
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The integral of a constant times a function is the constant times the integral of the function:
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Don't know the steps in finding this integral.
But the integral is
So, the result is:
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The result is:
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Now simplify:
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Add the constant of integration:
The answer is:
The answer (Indefinite)
[src]
/ | / / 2 3\\ / / 2 3\\ | x - 6 | 4 2 |729 54045*t 19127*t 486405*t || | 4 2 | 227152 4919296*t 776646*t 8301312*t || | ------------ dx = C - 6*RootSum|96080*t + 648*t + 48*t + 1, t -> t*log|--- + x - -------- + ------- + ---------|| + RootSum|24020*t + 512*t + 9*t + 2, t -> t*log|- ------ + x - ---------- + -------- + ----------|| | 4 \ \512 128 256 64 // \ \ 104865 20973 34955 6991 // | x + 6*x + 8 | /
$$\int \frac{x - 6}{\left(x^{4} + 6 x\right) + 8}\, dx = C + \operatorname{RootSum} {\left(24020 t^{4} + 512 t^{2} + 9 t + 2, \left( t \mapsto t \log{\left(\frac{8301312 t^{3}}{6991} - \frac{4919296 t^{2}}{20973} + \frac{776646 t}{34955} + x - \frac{227152}{104865} \right)} \right)\right)} - 6 \operatorname{RootSum} {\left(96080 t^{4} + 648 t^{2} + 48 t + 1, \left( t \mapsto t \log{\left(\frac{486405 t^{3}}{64} - \frac{54045 t^{2}}{128} + \frac{19127 t}{256} + x + \frac{729}{512} \right)} \right)\right)}$$
The graph
The answer
[src]
/ / 3 2\\ / / 3 2\\
| 4 2 |23465078 300499808*t 134508634*t 56764064*t || | 4 2 |25116929 300499808*t 134508634*t 56764064*t ||
- RootSum|4804*t + 1960*t - 747*t + 67, t -> t*log|-------- - ------------ - ----------- - -----------|| + RootSum|4804*t + 1960*t - 747*t + 67, t -> t*log|-------- - ------------ - ----------- - -----------||
\ \1651851 1651851 1651851 1651851 // \ \1651851 1651851 1651851 1651851 //
$$- \operatorname{RootSum} {\left(4804 t^{4} + 1960 t^{2} - 747 t + 67, \left( t \mapsto t \log{\left(- \frac{300499808 t^{3}}{1651851} - \frac{56764064 t^{2}}{1651851} - \frac{134508634 t}{1651851} + \frac{23465078}{1651851} \right)} \right)\right)} + \operatorname{RootSum} {\left(4804 t^{4} + 1960 t^{2} - 747 t + 67, \left( t \mapsto t \log{\left(- \frac{300499808 t^{3}}{1651851} - \frac{56764064 t^{2}}{1651851} - \frac{134508634 t}{1651851} + \frac{25116929}{1651851} \right)} \right)\right)}$$
=
=
/ / 3 2\\ / / 3 2\\
| 4 2 |23465078 300499808*t 134508634*t 56764064*t || | 4 2 |25116929 300499808*t 134508634*t 56764064*t ||
- RootSum|4804*t + 1960*t - 747*t + 67, t -> t*log|-------- - ------------ - ----------- - -----------|| + RootSum|4804*t + 1960*t - 747*t + 67, t -> t*log|-------- - ------------ - ----------- - -----------||
\ \1651851 1651851 1651851 1651851 // \ \1651851 1651851 1651851 1651851 //
$$- \operatorname{RootSum} {\left(4804 t^{4} + 1960 t^{2} - 747 t + 67, \left( t \mapsto t \log{\left(- \frac{300499808 t^{3}}{1651851} - \frac{56764064 t^{2}}{1651851} - \frac{134508634 t}{1651851} + \frac{23465078}{1651851} \right)} \right)\right)} + \operatorname{RootSum} {\left(4804 t^{4} + 1960 t^{2} - 747 t + 67, \left( t \mapsto t \log{\left(- \frac{300499808 t^{3}}{1651851} - \frac{56764064 t^{2}}{1651851} - \frac{134508634 t}{1651851} + \frac{25116929}{1651851} \right)} \right)\right)}$$
-RootSum(4804*_t^4 + 1960*_t^2 - 747*_t + 67, Lambda(_t, _t*log(23465078/1651851 - 300499808*_t^3/1651851 - 134508634*_t/1651851 - 56764064*_t^2/1651851))) + RootSum(4804*_t^4 + 1960*_t^2 - 747*_t + 67, Lambda(_t, _t*log(25116929/1651851 - 300499808*_t^3/1651851 - 134508634*_t/1651851 - 56764064*_t^2/1651851)))
Use the examples entering the upper and lower limits of integration.