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How to use it?
- Integral of d{x}:
-
Integral of ex^2
-
Integral of 1-7*x^2
-
Integral of xe^(1-x)
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Integral of -xe^x
-
Identical expressions
- (one -x)/(one +x²)
- (1 minus x) divide by (1 plus x²)
- (one minus x) divide by (one plus x²)
- 1-x/1+x²
- (1-x) divide by (1+x²)
- (1-x)/(1+x²)dx
-
Similar expressions
- (1-x)/(1-x²)
- (1+x)/(1+x²)
Integral of (1-x)/(1+x²) dx
The solution
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[src]
1 / | | 1 - x | ------ dx | 2 | 1 + x | / 0
$$\int\limits_{0}^{1} \frac{1 - x}{x^{2} + 1}\, dx$$
Integral((1 - x)/(1 + x^2), (x, 0, 1))
Detail solution
We have the integral:
/ | | 1 - x | ------ dx | 2 | 1 + x | /
Rewrite the integrand
/ 2*x \
|------------|
| 2 |
1 - x \x + 0*x + 1/ 1
------ = - -------------- + -------------
2 2 / 2 \
1 + x 1*\(-x) + 1/or
/ | | 1 - x | ------ dx | 2 = | 1 + x | /
/
|
| 2*x
| ------------ dx
| 2
| x + 0*x + 1 /
| |
/ | 1
- ------------------ + | --------- dx
2 | 2
| (-x) + 1
|
/ In the integral
/
|
| 2*x
- | ------------ dx
| 2
| x + 0*x + 1
|
/
--------------------
2 do replacement
2 u = x
then
the integral =
/
|
| 1
- | ----- du
| 1 + u
|
/ -log(1 + u)
------------- = ------------
2 2 do backward replacement
/
|
| 2*x
- | ------------ dx
| 2
| x + 0*x + 1
| / 2\
/ -log\1 + x /
-------------------- = -------------
2 2 In the integral
/ | | 1 | --------- dx | 2 | (-x) + 1 | /
do replacement
v = -x
then
the integral =
/ | | 1 | ------ dv = atan(v) | 2 | 1 + v | /
do backward replacement
/ | | 1 | --------- dx = atan(x) | 2 | (-x) + 1 | /
Solution is:
/ 2\
log\1 + x /
C - ----------- + atan(x)
2
The answer (Indefinite)
[src]
/ | / 2\ | 1 - x log\1 + x / | ------ dx = C - ----------- + atan(x) | 2 2 | 1 + x | /
$$\int \frac{1 - x}{x^{2} + 1}\, dx = C - \frac{\log{\left(x^{2} + 1 \right)}}{2} + \operatorname{atan}{\left(x \right)}$$
The graph
The answer
[src]
log(2) pi
- ------ + --
2 4
$$- \frac{\log{\left(2 \right)}}{2} + \frac{\pi}{4}$$
=
=
log(2) pi
- ------ + --
2 4
$$- \frac{\log{\left(2 \right)}}{2} + \frac{\pi}{4}$$
-log(2)/2 + pi/4
The graph
Use the examples entering the upper and lower limits of integration.