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^3/(x^2+1)^2
-
Integral of 4*x^3
- Integral of ln(cosx)
-
Integral of 2x*sinx
-
Identical expressions
- one /(sin^2x+2cos^2x)
- 1 divide by ( sinus of squared x plus 2 co sinus of e of squared x)
- one divide by ( sinus of squared x plus 2 co sinus of e of squared x)
- 1/(sin2x+2cos2x)
- 1/sin2x+2cos2x
- 1/(sin²x+2cos²x)
- 1/(sin to the power of 2x+2cos to the power of 2x)
- 1/sin^2x+2cos^2x
- 1 divide by (sin^2x+2cos^2x)
- 1/(sin^2x+2cos^2x)dx
-
Similar expressions
- 1/(sin^2x-2cos^2x)
Integral of 1/(sin^2x+2cos^2x) dx
The solution
You have entered
[src]
1 / | | 1 | ------------------- dx | 2 2 | sin (x) + 2*cos (x) | / 0
$$\int\limits_{0}^{1} \frac{1}{\sin^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}}\, dx$$
Integral(1/(sin(x)^2 + 2*cos(x)^2), (x, 0, 1))
The answer (Indefinite)
[src]
/ /x pi\ \ / /x pi\ \
| |- - --| | | |- - --| |
/ ___ | |2 2 | / ___ /x\\| ___ | |2 2 | / ___ /x\\|
| \/ 2 *|pi*floor|------| + atan|1 + \/ 2 *tan|-||| \/ 2 *|pi*floor|------| + atan|-1 + \/ 2 *tan|-|||
| 1 \ \ pi / \ \2/// \ \ pi / \ \2///
| ------------------- dx = C + ------------------------------------------------- + --------------------------------------------------
| 2 2 2 2
| sin (x) + 2*cos (x)
|
/
$$\int \frac{1}{\sin^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}}\, dx = C + \frac{\sqrt{2} \left(\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} - 1 \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{2} + \frac{\sqrt{2} \left(\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} + 1 \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{2}$$
The graph
The answer
[src]
___ / / ___ \\ ___ / / ___ \\
___ \/ 2 *\-pi - atan\1 - \/ 2 *tan(1/2)// \/ 2 *\-pi + atan\1 + \/ 2 *tan(1/2)//
pi*\/ 2 + -------------------------------------- + --------------------------------------
2 2
$$\frac{\sqrt{2} \left(- \pi - \operatorname{atan}{\left(- \sqrt{2} \tan{\left(\frac{1}{2} \right)} + 1 \right)}\right)}{2} + \frac{\sqrt{2} \left(- \pi + \operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{1}{2} \right)} + 1 \right)}\right)}{2} + \sqrt{2} \pi$$
=
=
___ / / ___ \\ ___ / / ___ \\
___ \/ 2 *\-pi - atan\1 - \/ 2 *tan(1/2)// \/ 2 *\-pi + atan\1 + \/ 2 *tan(1/2)//
pi*\/ 2 + -------------------------------------- + --------------------------------------
2 2
$$\frac{\sqrt{2} \left(- \pi - \operatorname{atan}{\left(- \sqrt{2} \tan{\left(\frac{1}{2} \right)} + 1 \right)}\right)}{2} + \frac{\sqrt{2} \left(- \pi + \operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{1}{2} \right)} + 1 \right)}\right)}{2} + \sqrt{2} \pi$$
pi*sqrt(2) + sqrt(2)*(-pi - atan(1 - sqrt(2)*tan(1/2)))/2 + sqrt(2)*(-pi + atan(1 + sqrt(2)*tan(1/2)))/2
The graph
Use the examples entering the upper and lower limits of integration.