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How to use it?
- Integral of d{x}:
-
Integral of 1/sqrt(1-x)
-
Integral of (x^2)^(1/3)
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Integral of (x+3)
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Integral of (x+1)dx
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Identical expressions
- (x)/(x^ two +3x+ four)
- (x) divide by (x squared plus 3x plus 4)
- (x) divide by (x to the power of two plus 3x plus four)
- (x)/(x2+3x+4)
- x/x2+3x+4
- (x)/(x²+3x+4)
- (x)/(x to the power of 2+3x+4)
- x/x^2+3x+4
- (x) divide by (x^2+3x+4)
- (x)/(x^2+3x+4)dx
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Similar expressions
- (x)/(x^2+3x-4)
- (x)/(x^2-3x+4)
Integral of (x)/(x^2+3x+4) dx
The solution
You have entered
[src]
1 / | | x | ------------ dx | 2 | x + 3*x + 4 | / 0
$$\int\limits_{0}^{1} \frac{x}{x^{2} + 3 x + 4}\, dx$$
Integral(x/(x^2 + 3*x + 4), (x, 0, 1))
Detail solution
We have the integral:
/ | | x | 1*------------ dx | 2 | x + 3*x + 4 | /
Rewrite the integrand
/ 1*2*x + 3 \
|--------------| / -3 \
| 2 | |-----|
x \1*x + 3*x + 4/ \2*7/4/
------------ = ---------------- + ---------------------------
2 2 2
x + 3*x + 4 / ___ ___\
|-2*\/ 7 3*\/ 7 |
|--------*x - -------| + 1
\ 7 7 / or
/ | | x | 1*------------ dx | 2 = | x + 3*x + 4 | /
/
|
| 1
/ 6* | --------------------------- dx
| | 2
| 1*2*x + 3 | / ___ ___\
| -------------- dx | |-2*\/ 7 3*\/ 7 |
| 2 | |--------*x - -------| + 1
| 1*x + 3*x + 4 | \ 7 7 /
| |
/ /
-------------------- - -----------------------------------
2 7 In the integral
/
|
| 1*2*x + 3
| -------------- dx
| 2
| 1*x + 3*x + 4
|
/
--------------------
2 do replacement
2 u = x + 3*x
then
the integral =
/
|
| 1
| ----- du
| 4 + u
|
/ log(4 + u)
----------- = ----------
2 2 do backward replacement
/
|
| 1*2*x + 3
| -------------- dx
| 2
| 1*x + 3*x + 4
| / 2 \
/ log\4 + x + 3*x/
-------------------- = -----------------
2 2 In the integral
/
|
| 1
-6* | --------------------------- dx
| 2
| / ___ ___\
| |-2*\/ 7 3*\/ 7 |
| |--------*x - -------| + 1
| \ 7 7 /
|
/
------------------------------------
7 do replacement
___ ___
3*\/ 7 2*x*\/ 7
v = - ------- - ---------
7 7 then
the integral =
/
|
| 1
-6* | ------ dv
| 2
| 1 + v
|
/ -6*atan(v)
--------------- = ----------
7 7 do backward replacement
/
|
| 1
-6* | --------------------------- dx
| 2
| / ___ ___\
| |-2*\/ 7 3*\/ 7 |
| |--------*x - -------| + 1 / ___ ___\
| \ 7 7 / ___ |3*\/ 7 2*x*\/ 7 |
| -3*\/ 7 *atan|------- + ---------|
/ \ 7 7 /
------------------------------------ = ----------------------------------
7 7 Solution is:
/ ___ ___\
___ |3*\/ 7 2*x*\/ 7 |
/ 2 \ 3*\/ 7 *atan|------- + ---------|
log\4 + x + 3*x/ \ 7 7 /
C + ----------------- - ---------------------------------
2 7
The answer (Indefinite)
[src]
/ ___ \ / ___ |2*\/ 7 *(3/2 + x)| | / 2 \ 3*\/ 7 *atan|-----------------| | x log\4 + x + 3*x/ \ 7 / | ------------ dx = C + ----------------- - ------------------------------- | 2 2 7 | x + 3*x + 4 | /
$${{\log \left(x^2+3\,x+4\right)}\over{2}}-{{3\,\arctan \left({{2\,x+
3}\over{\sqrt{7}}}\right)}\over{\sqrt{7}}}$$
The graph
The answer
[src]
/ ___\ / ___\
___ |5*\/ 7 | ___ |3*\/ 7 |
3*\/ 7 *atan|-------| 3*\/ 7 *atan|-------|
log(8) log(4) \ 7 / \ 7 /
------ - ------ - --------------------- + ---------------------
2 2 7 7
$$-{{3\,\arctan \left({{5}\over{\sqrt{7}}}\right)}\over{\sqrt{7}}}+{{
3\,\arctan \left({{3}\over{\sqrt{7}}}\right)}\over{\sqrt{7}}}+{{
\log 8}\over{2}}-{{\log 4}\over{2}}$$
=
=
/ ___\ / ___\
___ |5*\/ 7 | ___ |3*\/ 7 |
3*\/ 7 *atan|-------| 3*\/ 7 *atan|-------|
log(8) log(4) \ 7 / \ 7 /
------ - ------ - --------------------- + ---------------------
2 2 7 7
$$- \frac{3 \sqrt{7} \operatorname{atan}{\left(\frac{5 \sqrt{7}}{7} \right)}}{7} - \frac{\log{\left(4 \right)}}{2} + \frac{3 \sqrt{7} \operatorname{atan}{\left(\frac{3 \sqrt{7}}{7} \right)}}{7} + \frac{\log{\left(8 \right)}}{2}$$
The graph
Use the examples entering the upper and lower limits of integration.