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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(1+u^2)
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Integral of x*sin2x
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Integral of (1/x^2)dx
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Integral of -x^3
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Identical expressions
- one /(x*(sqrt(two)*(x))- nine)
- 1 divide by (x multiply by ( square root of (2) multiply by (x)) minus 9)
- one divide by (x multiply by ( square root of (two) multiply by (x)) minus nine)
- 1/(x*(√(2)*(x))-9)
- 1/(x(sqrt(2)(x))-9)
- 1/xsqrt2x-9
- 1 divide by (x*(sqrt(2)*(x))-9)
- 1/(x*(sqrt(2)*(x))-9)dx
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Similar expressions
- 1/(x*(sqrt(2)*(x))+9)
Integral of 1/(x*(sqrt(2)*(x))-9) dx
The solution
You have entered
[src]
1 / | | 1 | ------------- dx | ___ | x*\/ 2 *x - 9 | / 0
$$\int\limits_{0}^{1} \frac{1}{x \sqrt{2} x - 9}\, dx$$
Integral(1/(x*(sqrt(2)*x) - 9), (x, 0, 1))
Detail solution
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Add the constant of integration:
PiecewiseRule(subfunctions=[(ArctanRule(a=1, b=sqrt(2), c=-9, context=1/(x*(sqrt(2)*x) - 9), symbol=x), False), (ArccothRule(a=1, b=sqrt(2), c=-9, context=1/(x*(sqrt(2)*x) - 9), symbol=x), x**2 > 9*sqrt(2)/2), (ArctanhRule(a=1, b=sqrt(2), c=-9, context=1/(x*(sqrt(2)*x) - 9), symbol=x), x**2 < 9*sqrt(2)/2)], context=1/(x*(sqrt(2)*x) - 9), symbol=x)
The answer is:
The answer (Indefinite)
[src]
// / 4 ___\ \
|| 3/4 |x*\/ 2 | |
||-2 *acoth|-------| ___|
/ || \ 3 / 2 9*\/ 2 |
| ||--------------------- for x > -------|
| 1 || 6 2 |
| ------------- dx = C + |< |
| ___ || / 4 ___\ |
| x*\/ 2 *x - 9 || 3/4 |x*\/ 2 | |
| ||-2 *atanh|-------| ___|
/ || \ 3 / 2 9*\/ 2 |
||--------------------- for x < -------|
\\ 6 2 /
$$\int \frac{1}{x \sqrt{2} x - 9}\, dx = C + \begin{cases} - \frac{2^{\frac{3}{4}} \operatorname{acoth}{\left(\frac{\sqrt[4]{2} x}{3} \right)}}{6} & \text{for}\: x^{2} > \frac{9 \sqrt{2}}{2} \\- \frac{2^{\frac{3}{4}} \operatorname{atanh}{\left(\frac{\sqrt[4]{2} x}{3} \right)}}{6} & \text{for}\: x^{2} < \frac{9 \sqrt{2}}{2} \end{cases}$$
The graph
The answer
[src]
/ / 3/4\\ / 3/4\ / / 3/4\\ / 3/4\
3/4 | |3*2 || 3/4 | 3*2 | 3/4 | | 3*2 || 3/4 |3*2 |
2 *|pi*I + log|------|| 2 *log|1 + ------| 2 *|pi*I + log|-1 + ------|| 2 *log|------|
\ \ 2 // \ 2 / \ \ 2 // \ 2 /
- ------------------------- - -------------------- + ------------------------------ + ----------------
12 12 12 12
$$- \frac{2^{\frac{3}{4}} \log{\left(1 + \frac{3 \cdot 2^{\frac{3}{4}}}{2} \right)}}{12} + \frac{2^{\frac{3}{4}} \log{\left(\frac{3 \cdot 2^{\frac{3}{4}}}{2} \right)}}{12} - \frac{2^{\frac{3}{4}} \left(\log{\left(\frac{3 \cdot 2^{\frac{3}{4}}}{2} \right)} + i \pi\right)}{12} + \frac{2^{\frac{3}{4}} \left(\log{\left(-1 + \frac{3 \cdot 2^{\frac{3}{4}}}{2} \right)} + i \pi\right)}{12}$$
=
=
/ / 3/4\\ / 3/4\ / / 3/4\\ / 3/4\
3/4 | |3*2 || 3/4 | 3*2 | 3/4 | | 3*2 || 3/4 |3*2 |
2 *|pi*I + log|------|| 2 *log|1 + ------| 2 *|pi*I + log|-1 + ------|| 2 *log|------|
\ \ 2 // \ 2 / \ \ 2 // \ 2 /
- ------------------------- - -------------------- + ------------------------------ + ----------------
12 12 12 12
$$- \frac{2^{\frac{3}{4}} \log{\left(1 + \frac{3 \cdot 2^{\frac{3}{4}}}{2} \right)}}{12} + \frac{2^{\frac{3}{4}} \log{\left(\frac{3 \cdot 2^{\frac{3}{4}}}{2} \right)}}{12} - \frac{2^{\frac{3}{4}} \left(\log{\left(\frac{3 \cdot 2^{\frac{3}{4}}}{2} \right)} + i \pi\right)}{12} + \frac{2^{\frac{3}{4}} \left(\log{\left(-1 + \frac{3 \cdot 2^{\frac{3}{4}}}{2} \right)} + i \pi\right)}{12}$$
-2^(3/4)*(pi*i + log(3*2^(3/4)/2))/12 - 2^(3/4)*log(1 + 3*2^(3/4)/2)/12 + 2^(3/4)*(pi*i + log(-1 + 3*2^(3/4)/2))/12 + 2^(3/4)*log(3*2^(3/4)/2)/12
Use the examples entering the upper and lower limits of integration.