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How to use it?
- Integral of d{x}:
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Integral of xsqrt(x²+1)
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Integral of (x+3)/(x^2-5x+6)
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Integral of exp(-cos(x))*sin(x)
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Integral of e^((-x)^2)*x*dx
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Identical expressions
- dx/cos
- dx divide by co sinus of e of
- dx divide by cos
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Similar expressions
- ((e^tan(x))*dx)/cos(x)^2
- (-sin(x)*dx)/cos(x)
Integral of dx/cos dx
The solution
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[src]
1 / | | 1 | ------ dx | cos(x) | / 0
$$\int\limits_{0}^{1} \frac{1}{\cos{\left(x \right)}}\, dx$$
Integral(1/cos(x), (x, 0, 1))
Detail solution
We have the integral:
/ | | 1 | ------ dx | cos(x) | /
The integrand
1 ------ cos(x)
Multiply numerator and denominator by
cos(x)
we get
1 cos(x)
------ = -------
cos(x) 2
cos (x)Because
sin(a)^2 + cos(a)^2 = 1
then
2 2 cos (x) = 1 - sin (x)
transform the denominator
cos(x) cos(x) ------- = ----------- 2 2 cos (x) 1 - sin (x)
do replacement
u = sin(x)
then the integral
/ | | cos(x) | ----------- dx | 2 = | 1 - sin (x) | /
/ | | cos(x) | ----------- dx | 2 = | 1 - sin (x) | /
Because du = dx*cos(x)
/ | | 1 | ------ du | 2 | 1 - u | /
Rewrite the integrand
1 1
----- + -----
1 1 - u 1 + u
------ = -------------
2 2
1 - u then
/ /
| |
| 1 | 1
| ----- du | ----- du
/ | 1 + u | 1 - u
| | |
| 1 / / =
| ------ du = ----------- + -----------
| 2 2 2
| 1 - u
|
/
= log(1 + u)/2 - log(-1 + u)/2
do backward replacement
u = sin(x)
The answer
/ | | 1 log(1 + sin(x)) log(-1 + sin(x)) | ------ dx = --------------- - ---------------- + C0 | cos(x) 2 2 | /
where C0 is constant, independent of x
The answer (Indefinite)
[src]
/ | | 1 log(1 + sin(x)) log(-1 + sin(x)) | ------ dx = C + --------------- - ---------------- | cos(x) 2 2 | /
$$\int \frac{1}{\cos{\left(x \right)}}\, dx = C - \frac{\log{\left(\sin{\left(x \right)} - 1 \right)}}{2} + \frac{\log{\left(\sin{\left(x \right)} + 1 \right)}}{2}$$
The graph
The answer
[src]
log(1 + sin(1)) log(1 - sin(1))
--------------- - ---------------
2 2
$$\frac{\log{\left(\sin{\left(1 \right)} + 1 \right)}}{2} - \frac{\log{\left(1 - \sin{\left(1 \right)} \right)}}{2}$$
=
=
log(1 + sin(1)) log(1 - sin(1))
--------------- - ---------------
2 2
$$\frac{\log{\left(\sin{\left(1 \right)} + 1 \right)}}{2} - \frac{\log{\left(1 - \sin{\left(1 \right)} \right)}}{2}$$
log(1 + sin(1))/2 - log(1 - sin(1))/2
The graph
Use the examples entering the upper and lower limits of integration.