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How to use it?
- Integral of d{x}:
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Integral of (-1+u)/(1+u^2)
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Integral of cos2xsinx
- Integral of √(1-cos(x))
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Integral of 1/(1+x^(1/2))
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Identical expressions
- (- one +u)/(one +u^ two)
- ( minus 1 plus u) divide by (1 plus u squared )
- ( minus one plus u) divide by (one plus u to the power of two)
- (-1+u)/(1+u2)
- -1+u/1+u2
- (-1+u)/(1+u²)
- (-1+u)/(1+u to the power of 2)
- -1+u/1+u^2
- (-1+u) divide by (1+u^2)
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Similar expressions
- (1+u)/(1+u^2)
- (-1-u)/(1+u^2)
- (-1+u)/(1-u^2)
Integral of (-1+u)/(1+u^2) du
The solution
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[src]
1 / | | -1 + u | ------ du | 2 | 1 + u | / 0
$$\int\limits_{0}^{1} \frac{u - 1}{u^{2} + 1}\, du$$
Integral((-1 + u)/(1 + u^2), (u, 0, 1))
Detail solution
We have the integral:
/ | | -1 + u | ------ du | 2 | 1 + u | /
Rewrite the integrand
/ 2*u \
|------------| /-1 \
| 2 | |---|
-1 + u \u + 0*u + 1/ \ 1 /
------ = -------------- + ---------
2 2 2
1 + u (-u) + 1or
/ | | -1 + u | ------ du | 2 = | 1 + u | /
/
|
| 2*u
| ------------ du
| 2
| u + 0*u + 1 /
| |
/ | 1
------------------ - | --------- du
2 | 2
| (-u) + 1
|
/ In the integral
/
|
| 2*u
| ------------ du
| 2
| u + 0*u + 1
|
/
------------------
2 do replacement
2 u = u
then
the integral =
/
|
| 1
| ----- du
| 1 + u
|
/ log(1 + u)
----------- = ----------
2 2 do backward replacement
/
|
| 2*u
| ------------ du
| 2
| u + 0*u + 1
| / 2\
/ log\1 + u /
------------------ = -----------
2 2 In the integral
/ | | 1 - | --------- du | 2 | (-u) + 1 | /
do replacement
v = -u
then
the integral =
/ | | 1 - | ------ dv = -atan(v) | 2 | 1 + v | /
do backward replacement
/ | | 1 - | --------- du = -atan(u) | 2 | (-u) + 1 | /
Solution is:
/ 2\
log\1 + u /
C + ----------- - atan(u)
2
The answer (Indefinite)
[src]
/ | / 2\ | -1 + u log\1 + u / | ------ du = C + ----------- - atan(u) | 2 2 | 1 + u | /
$$\int \frac{u - 1}{u^{2} + 1}\, du = C + \frac{\log{\left(u^{2} + 1 \right)}}{2} - \operatorname{atan}{\left(u \right)}$$
The graph
The answer
[src]
log(2) pi ------ - -- 2 4
$$- \frac{\pi}{4} + \frac{\log{\left(2 \right)}}{2}$$
=
=
log(2) pi ------ - -- 2 4
$$- \frac{\pi}{4} + \frac{\log{\left(2 \right)}}{2}$$
log(2)/2 - pi/4
The graph
Use the examples entering the upper and lower limits of integration.