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- Improper integral
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How to use it?
- Integral of d{x}:
-
Integral of sin(x)^2
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Integral of ln
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Integral of e^2
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Integral of cos^3(3t)sin(3t)dt
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Identical expressions
- (xdx)/((x^ two)+7x+ thirteen)
- (xdx) divide by ((x squared ) plus 7x plus 13)
- (xdx) divide by ((x to the power of two) plus 7x plus thirteen)
- (xdx)/((x2)+7x+13)
- xdx/x2+7x+13
- (xdx)/((x²)+7x+13)
- (xdx)/((x to the power of 2)+7x+13)
- xdx/x^2+7x+13
- (xdx) divide by ((x^2)+7x+13)
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Similar expressions
- (xdx)/((x^2)-7x+13)
- (xdx)/((x^2)+7x-13)
Integral of (xdx)/((x^2)+7x+13) dx
The solution
You have entered
[src]
1 / | | 1 | x*1*------------- dx | 2 | x + 7*x + 13 | / 0
$$\int\limits_{0}^{1} x 1 \cdot \frac{1}{x^{2} + 7 x + 13}\, dx$$
Detail solution
We have the integral:
/ | | 1 | 1*x*1*------------- dx | 2 | x + 7*x + 13 | /
Rewrite the integrand
/ 1*2*x + 7 \
|---------------| / -7 \
| 2 | |-----|
1 \1*x + 7*x + 13/ \2*3/4/
x*1*------------- = ----------------- + ---------------------------
2 2 2
x + 7*x + 13 / ___ ___\
|-2*\/ 3 7*\/ 3 |
|--------*x - -------| + 1
\ 3 3 / or
/ | | 1 | 1*x*1*------------- dx | 2 = | x + 7*x + 13 | /
/
|
| 1
/ 14* | --------------------------- dx
| | 2
| 1*2*x + 7 | / ___ ___\
| --------------- dx | |-2*\/ 3 7*\/ 3 |
| 2 | |--------*x - -------| + 1
| 1*x + 7*x + 13 | \ 3 3 /
| |
/ /
--------------------- - ------------------------------------
2 3 In the integral
/
|
| 1*2*x + 7
| --------------- dx
| 2
| 1*x + 7*x + 13
|
/
---------------------
2 do replacement
2 u = x + 7*x
then
the integral =
/
|
| 1
| ------ du
| 13 + u
|
/ log(13 + u)
------------ = -----------
2 2 do backward replacement
/
|
| 1*2*x + 7
| --------------- dx
| 2
| 1*x + 7*x + 13
| / 2 \
/ log\13 + x + 7*x/
--------------------- = ------------------
2 2 In the integral
/
|
| 1
-14* | --------------------------- dx
| 2
| / ___ ___\
| |-2*\/ 3 7*\/ 3 |
| |--------*x - -------| + 1
| \ 3 3 /
|
/
-------------------------------------
3 do replacement
___ ___
7*\/ 3 2*x*\/ 3
v = - ------- - ---------
3 3 then
the integral =
/
|
| 1
-14* | ------ dv
| 2
| 1 + v
|
/ -14*atan(v)
---------------- = -----------
3 3 do backward replacement
/
|
| 1
-14* | --------------------------- dx
| 2
| / ___ ___\
| |-2*\/ 3 7*\/ 3 |
| |--------*x - -------| + 1 / ___ ___\
| \ 3 3 / ___ |7*\/ 3 2*x*\/ 3 |
| -7*\/ 3 *atan|------- + ---------|
/ \ 3 3 /
------------------------------------- = ----------------------------------
3 3 Solution is:
/ ___ ___\
___ |7*\/ 3 2*x*\/ 3 |
/ 2 \ 7*\/ 3 *atan|------- + ---------|
log\13 + x + 7*x/ \ 3 3 /
C + ------------------ - ---------------------------------
2 3
The answer (Indefinite)
[src]
/ ___ \ / ___ |2*\/ 3 *(7/2 + x)| | / 2 \ 7*\/ 3 *atan|-----------------| | 1 log\13 + x + 7*x/ \ 3 / | x*1*------------- dx = C + ------------------ - ------------------------------- | 2 2 3 | x + 7*x + 13 | /
$${{\log \left(x^2+7\,x+13\right)}\over{2}}-{{7\,\arctan \left({{2\,x
+7}\over{\sqrt{3}}}\right)}\over{\sqrt{3}}}$$
The graph
The answer
[src]
/ ___\
___ |7*\/ 3 |
___ / ___\ 7*\/ 3 *atan|-------|
log(21) log(13) 7*\/ 3 *atan\3*\/ 3 / \ 3 /
------- - ------- - --------------------- + ---------------------
2 2 3 3
$$-{{7\,\arctan 3^{{{3}\over{2}}}}\over{\sqrt{3}}}+{{7\,\arctan
\left({{7}\over{\sqrt{3}}}\right)}\over{\sqrt{3}}}+{{\log 21}\over{2
}}-{{\log 13}\over{2}}$$
=
=
/ ___\
___ |7*\/ 3 |
___ / ___\ 7*\/ 3 *atan|-------|
log(21) log(13) 7*\/ 3 *atan\3*\/ 3 / \ 3 /
------- - ------- - --------------------- + ---------------------
2 2 3 3
$$- \frac{7 \sqrt{3} \operatorname{atan}{\left(3 \sqrt{3} \right)}}{3} - \frac{\log{\left(13 \right)}}{2} + \frac{\log{\left(21 \right)}}{2} + \frac{7 \sqrt{3} \operatorname{atan}{\left(\frac{7 \sqrt{3}}{3} \right)}}{3}$$
The graph
Use the examples entering the upper and lower limits of integration.