Mister Exam
Other calculators
- Improper integral
- Double Integral
- Triple Integral
- Curvilinear integral
- Derivative Step by Step
- Differential equations Step by Step
- Limits Step by Step
-
How to use it?
- Integral of d{x}:
- Integral of x^2/(x-1)
-
Integral of (x+2)^2
-
Integral of ln(x+2)
-
Integral of 1/sqrt(1+u^2)
-
Identical expressions
- one /cos2x
- 1 divide by co sinus of e of 2x
- one divide by co sinus of e of 2x
- 1 divide by cos2x
- 1/cos2xdx
-
Similar expressions
- 1/cos^2(x(sqrt1+tgx))
- 1/cos^2x
- 1/cos^2x-sin^2x
- (1/cos^2x-sinx)dx
- (1/sin^2x+1/cos^2x)
Integral of 1/cos2x dx
The solution
You have entered
[src]
1 / | | 1 | -------- dx | cos(2*x) | / 0
$$\int\limits_{0}^{1} \frac{1}{\cos{\left(2 x \right)}}\, dx$$
Integral(1/cos(2*x), (x, 0, 1))
Detail solution
We have the integral:
/ | | 1 | -------- dx | cos(2*x) | /
The integrand
1 -------- cos(2*x)
Multiply numerator and denominator by
cos(2*x)
we get
1 cos(2*x)
-------- = ---------
cos(2*x) 2
cos (2*x)Because
sin(a)^2 + cos(a)^2 = 1
then
2 2 cos (2*x) = 1 - sin (2*x)
transform the denominator
cos(2*x) cos(2*x) --------- = ------------- 2 2 cos (2*x) 1 - sin (2*x)
do replacement
u = sin(2*x)
then the integral
/ | | cos(2*x) | ------------- dx | 2 = | 1 - sin (2*x) | /
/ | | cos(2*x) | ------------- dx | 2 = | 1 - sin (2*x) | /
Because du = 2*dx*cos(2*x)
/ | | 1 | ---------- du | / 2\ | 2*\1 - u / | /
Rewrite the integrand
1 1 / 1 1 \ ---------- = ---*|----- + -----| / 2\ 2*2 \1 - u 1 + u/ 2*\1 - u /
then
/ /
| |
| 1 | 1
| ----- du | ----- du
/ | 1 + u | 1 - u
| | |
| 1 / / =
| ---------- du = ----------- + -----------
| / 2\ 4 4
| 2*\1 - u /
|
/
= -log(-1 + u)/4 + log(1 + u)/4
do backward replacement
u = sin(2*x)
The answer
/ | | 1 log(-1 + sin(2*x)) log(1 + sin(2*x)) | -------- dx = - ------------------ + ----------------- + C0 | cos(2*x) 4 4 | /
where C0 is constant, independent of x
The answer (Indefinite)
[src]
/ | | 1 log(-1 + sin(2*x)) log(1 + sin(2*x)) | -------- dx = C - ------------------ + ----------------- | cos(2*x) 4 4 | /
$$\int \frac{1}{\cos{\left(2 x \right)}}\, dx = C - \frac{\log{\left(\sin{\left(2 x \right)} - 1 \right)}}{4} + \frac{\log{\left(\sin{\left(2 x \right)} + 1 \right)}}{4}$$
The graph
The graph
Use the examples entering the upper and lower limits of integration.