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^4
-
Integral of (sin(x))^2
-
Integral of e^(x)
-
Integral of sin(logx)
-
Identical expressions
- cos^ two (two x-(pi/ four))-sin^2(2x-(pi/ four))
- co sinus of e of squared (2x minus ( Pi divide by 4)) minus sinus of squared (2x minus ( Pi divide by 4))
- co sinus of e of to the power of two (two x minus ( Pi divide by four)) minus sinus of squared (2x minus ( Pi divide by four))
- cos2(2x-(pi/4))-sin2(2x-(pi/4))
- cos22x-pi/4-sin22x-pi/4
- cos²(2x-(pi/4))-sin²(2x-(pi/4))
- cos to the power of 2(2x-(pi/4))-sin to the power of 2(2x-(pi/4))
- cos^22x-pi/4-sin^22x-pi/4
- cos^2(2x-(pi divide by 4))-sin^2(2x-(pi divide by 4))
- cos^2(2x-(pi/4))-sin^2(2x-(pi/4))dx
-
Similar expressions
- cos^2(2x-(pi/4))+sin^2(2x-(pi/4))
- cos^2(2x-(pi/4))-sin^2(2x+(pi/4))
- cos^2(2x+(pi/4))-sin^2(2x-(pi/4))
Integral of cos^2(2x-(pi/4))-sin^2(2x-(pi/4)) dx
The solution
You have entered
[src]
pi -- 24 / | | / 2/ pi\ 2/ pi\\ | |cos |2*x - --| - sin |2*x - --|| dx | \ \ 4 / \ 4 // | / pi -- 12
$$\int\limits_{\frac{\pi}{12}}^{\frac{\pi}{24}} \left(- \sin^{2}{\left(2 x - \frac{\pi}{4} \right)} + \cos^{2}{\left(2 x - \frac{\pi}{4} \right)}\right)\, dx$$
Integral(cos(2*x - pi/4)^2 - sin(2*x - pi/4)^2, (x, pi/12, pi/24))
Detail solution
-
Integrate term-by-term:
-
The integral of a constant times a function is the constant times the integral of the function:
-
Don't know the steps in finding this integral.
But the integral is
So, the result is:
-
-
Don't know the steps in finding this integral.
But the integral is
The result is:
-
-
Now simplify:
-
Add the constant of integration:
The answer is:
The answer (Indefinite)
[src]
/ / 3*pi\ 3/ 3*pi\ | 2*cot|x + ----| 2*cot |x + ----| | / 2/ pi\ 2/ pi\\ \ 8 / \ 8 / | |cos |2*x - --| - sin |2*x - --|| dx = C - --------------------------------------- + --------------------------------------- | \ \ 4 / \ 4 // 4/ 3*pi\ 2/ 3*pi\ 4/ 3*pi\ 2/ 3*pi\ | 2 + 2*cot |x + ----| + 4*cot |x + ----| 2 + 2*cot |x + ----| + 4*cot |x + ----| / \ 8 / \ 8 / \ 8 / \ 8 /
$$\int \left(- \sin^{2}{\left(2 x - \frac{\pi}{4} \right)} + \cos^{2}{\left(2 x - \frac{\pi}{4} \right)}\right)\, dx = C + \frac{2 \cot^{3}{\left(x + \frac{3 \pi}{8} \right)}}{2 \cot^{4}{\left(x + \frac{3 \pi}{8} \right)} + 4 \cot^{2}{\left(x + \frac{3 \pi}{8} \right)} + 2} - \frac{2 \cot{\left(x + \frac{3 \pi}{8} \right)}}{2 \cot^{4}{\left(x + \frac{3 \pi}{8} \right)} + 4 \cot^{2}{\left(x + \frac{3 \pi}{8} \right)} + 2}$$
The graph
The answer
[src]
/ ___ ___\ / ___ ___\
| \/ 2 \/ 6 | |\/ 2 \/ 6 |
___ |- ----- + -----|*|----- + -----|
\/ 3 \ 4 4 / \ 4 4 /
- ----- + ---------------------------------
8 2
$$- \frac{\sqrt{3}}{8} + \frac{\left(- \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right) \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)}{2}$$
=
=
/ ___ ___\ / ___ ___\
| \/ 2 \/ 6 | |\/ 2 \/ 6 |
___ |- ----- + -----|*|----- + -----|
\/ 3 \ 4 4 / \ 4 4 /
- ----- + ---------------------------------
8 2
$$- \frac{\sqrt{3}}{8} + \frac{\left(- \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right) \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)}{2}$$
-sqrt(3)/8 + (-sqrt(2)/4 + sqrt(6)/4)*(sqrt(2)/4 + sqrt(6)/4)/2
Use the examples entering the upper and lower limits of integration.