Mister Exam
Other calculators
- Improper integral
- Double Integral
- Triple Integral
- Curvilinear integral
- Derivative Step by Step
- Differential equations Step by Step
- Limits Step by Step
-
How to use it?
- Integral of d{x}:
-
Integral of 2*x/(-1+x)
-
Integral of (x+1)e^(2x)
-
Integral of (x+1)/(2x²+3x-4)
- Integral of ln(x+√(1+x^2))
- How do you in partial fractions?:
- (x-1)/(x^2-x-1)
-
Identical expressions
- (x- one)/(x^ two -x- one)
- (x minus 1) divide by (x squared minus x minus 1)
- (x minus one) divide by (x to the power of two minus x minus one)
- (x-1)/(x2-x-1)
- x-1/x2-x-1
- (x-1)/(x²-x-1)
- (x-1)/(x to the power of 2-x-1)
- x-1/x^2-x-1
- (x-1) divide by (x^2-x-1)
- (x-1)/(x^2-x-1)dx
-
Similar expressions
- (x-1)/(x^2-x+1)
- (x+1)/(x^2-x-1)
- (x-1)/(x^2+x-1)
Integral of (x-1)/(x^2-x-1) dx
The solution
You have entered
[src]
1 / | | x - 1 | ---------- dx | 2 | x - x - 1 | / 0
$$\int\limits_{0}^{1} \frac{x - 1}{\left(x^{2} - x\right) - 1}\, dx$$
Integral((x - 1)/(x^2 - x - 1), (x, 0, 1))
The answer (Indefinite)
[src]
// / ___ \ \
|| ___ |2*\/ 5 *(-1/2 + x)| |
||-\/ 5 *acoth|------------------| |
/ || \ 5 / 2 |
| / 2 \ ||--------------------------------- for (-1/2 + x) > 5/4|
| x - 1 log\-1 + x - x/ || 10 |
| ---------- dx = C + ---------------- - 2*|< |
| 2 2 || / ___ \ |
| x - x - 1 || ___ |2*\/ 5 *(-1/2 + x)| |
| ||-\/ 5 *atanh|------------------| |
/ || \ 5 / 2 |
||--------------------------------- for (-1/2 + x) < 5/4|
\\ 10 /
$$\int \frac{x - 1}{\left(x^{2} - x\right) - 1}\, dx = C - 2 \left(\begin{cases} - \frac{\sqrt{5} \operatorname{acoth}{\left(\frac{2 \sqrt{5} \left(x - \frac{1}{2}\right)}{5} \right)}}{10} & \text{for}\: \left(x - \frac{1}{2}\right)^{2} > \frac{5}{4} \\- \frac{\sqrt{5} \operatorname{atanh}{\left(\frac{2 \sqrt{5} \left(x - \frac{1}{2}\right)}{5} \right)}}{10} & \text{for}\: \left(x - \frac{1}{2}\right)^{2} < \frac{5}{4} \end{cases}\right) + \frac{\log{\left(x^{2} - x - 1 \right)}}{2}$$
The graph
The answer
[src]
/ ___\ / / ___\\ / ___\ / ___\ / ___\ / / ___\\ / ___\ / ___\ |1 \/ 5 | | | 1 \/ 5 || |1 \/ 5 | |1 \/ 5 | |1 \/ 5 | | |1 \/ 5 || |1 \/ 5 | | 1 \/ 5 | |- - -----|*|pi*I + log|- - + -----|| + |- + -----|*log|- + -----| - |- - -----|*|pi*I + log|- + -----|| - |- + -----|*log|- - + -----| \2 10 / \ \ 2 2 // \2 10 / \2 2 / \2 10 / \ \2 2 // \2 10 / \ 2 2 /
$$\left(\frac{\sqrt{5}}{10} + \frac{1}{2}\right) \log{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} \right)} - \left(\frac{\sqrt{5}}{10} + \frac{1}{2}\right) \log{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} \right)} - \left(\frac{1}{2} - \frac{\sqrt{5}}{10}\right) \left(\log{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} \right)} + i \pi\right) + \left(\frac{1}{2} - \frac{\sqrt{5}}{10}\right) \left(\log{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} \right)} + i \pi\right)$$
=
=
/ ___\ / / ___\\ / ___\ / ___\ / ___\ / / ___\\ / ___\ / ___\ |1 \/ 5 | | | 1 \/ 5 || |1 \/ 5 | |1 \/ 5 | |1 \/ 5 | | |1 \/ 5 || |1 \/ 5 | | 1 \/ 5 | |- - -----|*|pi*I + log|- - + -----|| + |- + -----|*log|- + -----| - |- - -----|*|pi*I + log|- + -----|| - |- + -----|*log|- - + -----| \2 10 / \ \ 2 2 // \2 10 / \2 2 / \2 10 / \ \2 2 // \2 10 / \ 2 2 /
$$\left(\frac{\sqrt{5}}{10} + \frac{1}{2}\right) \log{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} \right)} - \left(\frac{\sqrt{5}}{10} + \frac{1}{2}\right) \log{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} \right)} - \left(\frac{1}{2} - \frac{\sqrt{5}}{10}\right) \left(\log{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} \right)} + i \pi\right) + \left(\frac{1}{2} - \frac{\sqrt{5}}{10}\right) \left(\log{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} \right)} + i \pi\right)$$
(1/2 - sqrt(5)/10)*(pi*i + log(-1/2 + sqrt(5)/2)) + (1/2 + sqrt(5)/10)*log(1/2 + sqrt(5)/2) - (1/2 - sqrt(5)/10)*(pi*i + log(1/2 + sqrt(5)/2)) - (1/2 + sqrt(5)/10)*log(-1/2 + sqrt(5)/2)
The graph
Use the examples entering the upper and lower limits of integration.