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How to use it?
- Integral of d{x}:
-
Integral of 1/(1+4x^2)
-
Integral of dx/(4x-1)
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Integral of (2x+3)/(x^2-5x+6)
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Integral of 2x(x^2+1)
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Identical expressions
- y*x/(y^ two +x^ two)
- y multiply by x divide by (y squared plus x squared )
- y multiply by x divide by (y to the power of two plus x to the power of two)
- y*x/(y2+x2)
- y*x/y2+x2
- y*x/(y²+x²)
- y*x/(y to the power of 2+x to the power of 2)
- yx/(y^2+x^2)
- yx/(y2+x2)
- yx/y2+x2
- yx/y^2+x^2
- y*x divide by (y^2+x^2)
- y*x/(y^2+x^2)dx
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Similar expressions
- y*x/(y^2-x^2)
Integral of y*x/(y^2+x^2) dx
The solution
You have entered
[src]
1 / | | y*x | ------- dx | 2 2 | y + x | / 0
$$\int\limits_{0}^{1} \frac{x y}{x^{2} + y^{2}}\, dx$$
Integral((y*x)/(y^2 + x^2), (x, 0, 1))
Detail solution
We have the integral:
/ | | y*x | ------- dx | 2 2 | y + x | /
Rewrite the integrand
/0 \
|--|
| 2|
y*x y 2*x \y /
------- = -*------------- + ------------
2 2 2 2 2 2
y + x x + 0*x + y /-1 \
|---*x| + 1
\ y / or
/ | | y*x | ------- dx | 2 2 = | y + x | /
/
|
| 2*x
y* | ------------- dx
| 2 2
| x + 0*x + y
|
/
---------------------
2 In the integral
/
|
| 2*x
y* | ------------- dx
| 2 2
| x + 0*x + y
|
/
---------------------
2 do replacement
2 u = x
then
the integral =
/
|
| 1
y* | ------ du
| 2
| u + y
| / 2\
/ y*log\u + y /
-------------- = -------------
2 2 do backward replacement
/
|
| 2*x
y* | ------------- dx
| 2 2
| x + 0*x + y
| / 2 2\
/ y*log\x + y /
--------------------- = --------------
2 2 In the integral
0
do replacement
-x
v = ---
y then
the integral =
True
do backward replacement
True
Solution is:
/ 2 2\
y*log\x + y /
C + --------------
2
The answer (Indefinite)
[src]
/ | / 2 2\ | y*x y*log\x + y / | ------- dx = C + -------------- | 2 2 2 | y + x | /
$$\int \frac{x y}{x^{2} + y^{2}}\, dx = C + \frac{y \log{\left(x^{2} + y^{2} \right)}}{2}$$
The answer
[src]
/ 2\ / 2\
y*log\1 + y / y*log\y /
------------- - ---------
2 2
$$- \frac{y \log{\left(y^{2} \right)}}{2} + \frac{y \log{\left(y^{2} + 1 \right)}}{2}$$
=
=
/ 2\ / 2\
y*log\1 + y / y*log\y /
------------- - ---------
2 2
$$- \frac{y \log{\left(y^{2} \right)}}{2} + \frac{y \log{\left(y^{2} + 1 \right)}}{2}$$
y*log(1 + y^2)/2 - y*log(y^2)/2
Use the examples entering the upper and lower limits of integration.