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How to use it?
- Integral of d{x}:
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Integral of x^(-1/2)
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Integral of sin²xcos²x
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Integral of x*2^(-x)
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Integral of y^(-2/3)
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Identical expressions
- (6x- seven)/(x^ two +4x+ thirteen)
- (6x minus 7) divide by (x squared plus 4x plus 13)
- (6x minus seven) divide by (x to the power of two plus 4x plus thirteen)
- (6x-7)/(x2+4x+13)
- 6x-7/x2+4x+13
- (6x-7)/(x²+4x+13)
- (6x-7)/(x to the power of 2+4x+13)
- 6x-7/x^2+4x+13
- (6x-7) divide by (x^2+4x+13)
- (6x-7)/(x^2+4x+13)dx
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Similar expressions
- (6x-7)/(x^2-4x+13)
- (6x+7)/(x^2+4x+13)
- (6x-7)/(x^2+4x-13)
Integral of (6x-7)/(x^2+4x+13) dx
The solution
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[src]
1 / | | 6*x - 7 | ------------- dx | 2 | x + 4*x + 13 | / 0
$$\int\limits_{0}^{1} \frac{6 x - 7}{\left(x^{2} + 4 x\right) + 13}\, dx$$
Integral((6*x - 7)/(x^2 + 4*x + 13), (x, 0, 1))
Detail solution
We have the integral:
/ | | 6*x - 7 | ------------- dx | 2 | x + 4*x + 13 | /
Rewrite the integrand
/-19 \
|----|
6*x - 7 2*x + 4 \ 9 /
------------- = 3*------------- + --------------
2 2 2
x + 4*x + 13 x + 4*x + 13 / x 2\
|- - - -| + 1
\ 3 3/ or
/ | | 6*x - 7 | ------------- dx | 2 = | x + 4*x + 13 | /
/
|
| 1
19* | -------------- dx
| 2
| / x 2\
| |- - - -| + 1
/ | \ 3 3/
| |
| 2*x + 4 /
3* | ------------- dx - -----------------------
| 2 9
| x + 4*x + 13
|
/ In the integral
/ | | 2*x + 4 3* | ------------- dx | 2 | x + 4*x + 13 | /
do replacement
2 u = x + 4*x
then
the integral =
/ | | 1 3* | ------ du = 3*log(13 + u) | 13 + u | /
do backward replacement
/ | | 2*x + 4 / 2 \ 3* | ------------- dx = 3*log\13 + x + 4*x/ | 2 | x + 4*x + 13 | /
In the integral
/
|
| 1
-19* | -------------- dx
| 2
| / x 2\
| |- - - -| + 1
| \ 3 3/
|
/
------------------------
9 do replacement
2 x
v = - - - -
3 3then
the integral =
/
|
| 1
-19* | ------ dv
| 2
| 1 + v
|
/ -19*atan(v)
---------------- = -----------
9 9 do backward replacement
/
|
| 1
-19* | -------------- dx
| 2
| / x 2\
| |- - - -| + 1
| \ 3 3/ /2 x\
| -19*atan|- + -|
/ \3 3/
------------------------ = ---------------
9 3 Solution is:
/2 x\
19*atan|- + -|
/ 2 \ \3 3/
C + 3*log\13 + x + 4*x/ - --------------
3
The answer (Indefinite)
[src]
/ /2 x\ | 19*atan|- + -| | 6*x - 7 / 2 \ \3 3/ | ------------- dx = C + 3*log\13 + x + 4*x/ - -------------- | 2 3 | x + 4*x + 13 | /
$$\int \frac{6 x - 7}{\left(x^{2} + 4 x\right) + 13}\, dx = C + 3 \log{\left(x^{2} + 4 x + 13 \right)} - \frac{19 \operatorname{atan}{\left(\frac{x}{3} + \frac{2}{3} \right)}}{3}$$
The graph
The answer
[src]
19*pi 19*atan(2/3)
-3*log(13) + 3*log(18) - ----- + ------------
12 3
$$- 3 \log{\left(13 \right)} - \frac{19 \pi}{12} + \frac{19 \operatorname{atan}{\left(\frac{2}{3} \right)}}{3} + 3 \log{\left(18 \right)}$$
=
=
19*pi 19*atan(2/3)
-3*log(13) + 3*log(18) - ----- + ------------
12 3
$$- 3 \log{\left(13 \right)} - \frac{19 \pi}{12} + \frac{19 \operatorname{atan}{\left(\frac{2}{3} \right)}}{3} + 3 \log{\left(18 \right)}$$
-3*log(13) + 3*log(18) - 19*pi/12 + 19*atan(2/3)/3
The graph
Use the examples entering the upper and lower limits of integration.