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Identical expressions
- sin(two x)sqrt(a^ two sin^2(x)+b^2cos^2(x))
- sinus of (2x) square root of (a squared sinus of squared (x) plus b squared co sinus of e of squared (x))
- sinus of (two x) square root of (a to the power of two sinus of squared (x) plus b squared co sinus of e of squared (x))
- sin(2x)√(a^2sin^2(x)+b^2cos^2(x))
- sin(2x)sqrt(a2sin2(x)+b2cos2(x))
- sin2xsqrta2sin2x+b2cos2x
- sin(2x)sqrt(a²sin²(x)+b²cos²(x))
- sin(2x)sqrt(a to the power of 2sin to the power of 2(x)+b to the power of 2cos to the power of 2(x))
- sin2xsqrta^2sin^2x+b^2cos^2x
- sin(2x)sqrt(a^2sin^2(x)+b^2cos^2(x))dx
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Similar expressions
- sin(2x)sqrt(a^2sin^2(x)-b^2cos^2(x))
Integral of sin(2x)sqrt(a^2sin^2(x)+b^2cos^2(x)) dx
The solution
You have entered
[src]
pi -- 2 / | | _________________________ | / 2 2 2 2 | sin(2*x)*\/ a *sin (x) + b *cos (x) dx | / 0
$$\int\limits_{0}^{\frac{\pi}{2}} \sqrt{a^{2} \sin^{2}{\left(x \right)} + b^{2} \cos^{2}{\left(x \right)}} \sin{\left(2 x \right)}\, dx$$
Integral(sin(2*x)*sqrt(a^2*sin(x)^2 + b^2*cos(x)^2), (x, 0, pi/2))
The answer (Indefinite)
[src]
/// 0 for And(a = 0, b = 0) \
||| |
||| _________________________ |
||| / 2 2 2 2 2 |
||| -\/ b *cos (x) + b *sin (x) *cos (x) |
||| -------------------------------------- for Or(a = -b, a = b) |
||| 2 2 2 |
/ ||< for - 2*b + 2*a = 0|
| ||| _________________________ _________________________ |
| _________________________ ||| 2 / 2 2 2 2 2 2 / 2 2 2 2 2 |
| / 2 2 2 2 |||a *\/ a *sin (x) + b *cos (x) *sin (x) b *\/ a *sin (x) + b *cos (x) *cos (x) |
| sin(2*x)*\/ a *sin (x) + b *cos (x) dx = C + 2*|<|--------------------------------------- + --------------------------------------- otherwise |
| ||| 2 2 2 2 |
/ ||| - 3*b + 3*a - 3*b + 3*a |
||\ |
|| |
|| 3/2 |
|| / 2 2 2 2 \ |
|| 2*\a *sin (x) + b *cos (x)/ |
|| ------------------------------ otherwise |
|| / 2 2\ |
|| 3*\- 2*b + 2*a / |
\\ /
$$\int \sqrt{a^{2} \sin^{2}{\left(x \right)} + b^{2} \cos^{2}{\left(x \right)}} \sin{\left(2 x \right)}\, dx = C + 2 \left(\begin{cases} \begin{cases} 0 & \text{for}\: a = 0 \wedge b = 0 \\- \frac{\sqrt{b^{2} \sin^{2}{\left(x \right)} + b^{2} \cos^{2}{\left(x \right)}} \cos^{2}{\left(x \right)}}{2} & \text{for}\: a = - b \vee a = b \\\frac{a^{2} \sqrt{a^{2} \sin^{2}{\left(x \right)} + b^{2} \cos^{2}{\left(x \right)}} \sin^{2}{\left(x \right)}}{3 a^{2} - 3 b^{2}} + \frac{b^{2} \sqrt{a^{2} \sin^{2}{\left(x \right)} + b^{2} \cos^{2}{\left(x \right)}} \cos^{2}{\left(x \right)}}{3 a^{2} - 3 b^{2}} & \text{otherwise} \end{cases} & \text{for}\: 2 a^{2} - 2 b^{2} = 0 \\\frac{2 \left(a^{2} \sin^{2}{\left(x \right)} + b^{2} \cos^{2}{\left(x \right)}\right)^{\frac{3}{2}}}{3 \left(2 a^{2} - 2 b^{2}\right)} & \text{otherwise} \end{cases}\right)$$
The answer
[src]
/ / / 2 2 \ / 2 2 \ / 2 2 \ / 2 2 \\ | 0 for Or\And\a = 0, b = 0, - 2*b + 2*a = 0/, And\a = 0, a = b, b = 0, - 2*b + 2*a = 0/, And\a = 0, a = -b, b = 0, - 2*b + 2*a = 0/, And\a = 0, a = -b, a = b, b = 0, - 2*b + 2*a = 0// | | 3/2 3/2 | / 2\ / 2\ | 4*\b / 4*\a / |- ----------------- + ----------------- for Or(And(a = 0, a = -b), And(a = 0, a = b), And(a = 0, b = 0), And(a = -b, a = b), And(a = b, b = 0), And(a = -b, b = 0), And(a = 0, a = -b, a = b), And(a = 0, a = b, b = 0), And(a = 0, a = -b, b = 0), And(a = -b, a = b, b = 0), And(a = 0, a = -b, a = b, b = 0), a = 0, a = -b, a = b, b = 0) | / 2 2\ / 2 2\ | 3*\- 2*b + 2*a / 3*\- 2*b + 2*a / | | ____ | / 2 / / 2 2 \ / 2 2 \ / 2 2 \ / 2 2 \ / 2 2 \ / 2 2 \ / 2 2 \ / 2 2 \ / 2 2 \\ | \/ b for Or\And\a = -b, - 2*b + 2*a = 0/, And\a = b, - 2*b + 2*a = 0/, And\a = 0, a = -b, - 2*b + 2*a = 0/, And\a = 0, a = b, - 2*b + 2*a = 0/, And\a = -b, a = b, - 2*b + 2*a = 0/, And\a = b, b = 0, - 2*b + 2*a = 0/, And\a = -b, b = 0, - 2*b + 2*a = 0/, And\a = 0, a = -b, a = b, - 2*b + 2*a = 0/, And\a = -b, a = b, b = 0, - 2*b + 2*a = 0// < | ____ ____ | 2 / 2 2 / 2 | 2*b *\/ b 2*a *\/ a / / 2 2 \ / 2 2 \ 2 2 \ | - ------------- + ------------- for Or\And\a = 0, - 2*b + 2*a = 0/, And\b = 0, - 2*b + 2*a = 0/, - 2*b + 2*a = 0/ | 2 2 2 2 | - 3*b + 3*a - 3*b + 3*a | | 3/2 3/2 | / 2\ / 2\ | 4*\b / 4*\a / |- ----------------- + ----------------- otherwise | / 2 2\ / 2 2\ \ 3*\- 2*b + 2*a / 3*\- 2*b + 2*a /
$$\begin{cases} 0 & \text{for}\: \left(a = 0 \wedge b = 0 \wedge 2 a^{2} - 2 b^{2} = 0\right) \vee \left(a = 0 \wedge a = b \wedge b = 0 \wedge 2 a^{2} - 2 b^{2} = 0\right) \vee \left(a = 0 \wedge a = - b \wedge b = 0 \wedge 2 a^{2} - 2 b^{2} = 0\right) \vee \left(a = 0 \wedge a = - b \wedge a = b \wedge b = 0 \wedge 2 a^{2} - 2 b^{2} = 0\right) \\\frac{4 \left(a^{2}\right)^{\frac{3}{2}}}{3 \left(2 a^{2} - 2 b^{2}\right)} - \frac{4 \left(b^{2}\right)^{\frac{3}{2}}}{3 \left(2 a^{2} - 2 b^{2}\right)} & \text{for}\: \left(a = 0 \wedge a = - b\right) \vee \left(a = 0 \wedge a = b\right) \vee \left(a = 0 \wedge b = 0\right) \vee \left(a = - b \wedge a = b\right) \vee \left(a = b \wedge b = 0\right) \vee \left(a = - b \wedge b = 0\right) \vee \left(a = 0 \wedge a = - b \wedge a = b\right) \vee \left(a = 0 \wedge a = b \wedge b = 0\right) \vee \left(a = 0 \wedge a = - b \wedge b = 0\right) \vee \left(a = - b \wedge a = b \wedge b = 0\right) \vee \left(a = 0 \wedge a = - b \wedge a = b \wedge b = 0\right) \vee a = 0 \vee a = - b \vee a = b \vee b = 0 \\\sqrt{b^{2}} & \text{for}\: \left(a = - b \wedge 2 a^{2} - 2 b^{2} = 0\right) \vee \left(a = b \wedge 2 a^{2} - 2 b^{2} = 0\right) \vee \left(a = 0 \wedge a = - b \wedge 2 a^{2} - 2 b^{2} = 0\right) \vee \left(a = 0 \wedge a = b \wedge 2 a^{2} - 2 b^{2} = 0\right) \vee \left(a = - b \wedge a = b \wedge 2 a^{2} - 2 b^{2} = 0\right) \vee \left(a = b \wedge b = 0 \wedge 2 a^{2} - 2 b^{2} = 0\right) \vee \left(a = - b \wedge b = 0 \wedge 2 a^{2} - 2 b^{2} = 0\right) \vee \left(a = 0 \wedge a = - b \wedge a = b \wedge 2 a^{2} - 2 b^{2} = 0\right) \vee \left(a = - b \wedge a = b \wedge b = 0 \wedge 2 a^{2} - 2 b^{2} = 0\right) \\\frac{2 a^{2} \sqrt{a^{2}}}{3 a^{2} - 3 b^{2}} - \frac{2 b^{2} \sqrt{b^{2}}}{3 a^{2} - 3 b^{2}} & \text{for}\: \left(a = 0 \wedge 2 a^{2} - 2 b^{2} = 0\right) \vee \left(b = 0 \wedge 2 a^{2} - 2 b^{2} = 0\right) \vee 2 a^{2} - 2 b^{2} = 0 \\\frac{4 \left(a^{2}\right)^{\frac{3}{2}}}{3 \left(2 a^{2} - 2 b^{2}\right)} - \frac{4 \left(b^{2}\right)^{\frac{3}{2}}}{3 \left(2 a^{2} - 2 b^{2}\right)} & \text{otherwise} \end{cases}$$
=
=
/ / / 2 2 \ / 2 2 \ / 2 2 \ / 2 2 \\ | 0 for Or\And\a = 0, b = 0, - 2*b + 2*a = 0/, And\a = 0, a = b, b = 0, - 2*b + 2*a = 0/, And\a = 0, a = -b, b = 0, - 2*b + 2*a = 0/, And\a = 0, a = -b, a = b, b = 0, - 2*b + 2*a = 0// | | 3/2 3/2 | / 2\ / 2\ | 4*\b / 4*\a / |- ----------------- + ----------------- for Or(And(a = 0, a = -b), And(a = 0, a = b), And(a = 0, b = 0), And(a = -b, a = b), And(a = b, b = 0), And(a = -b, b = 0), And(a = 0, a = -b, a = b), And(a = 0, a = b, b = 0), And(a = 0, a = -b, b = 0), And(a = -b, a = b, b = 0), And(a = 0, a = -b, a = b, b = 0), a = 0, a = -b, a = b, b = 0) | / 2 2\ / 2 2\ | 3*\- 2*b + 2*a / 3*\- 2*b + 2*a / | | ____ | / 2 / / 2 2 \ / 2 2 \ / 2 2 \ / 2 2 \ / 2 2 \ / 2 2 \ / 2 2 \ / 2 2 \ / 2 2 \\ | \/ b for Or\And\a = -b, - 2*b + 2*a = 0/, And\a = b, - 2*b + 2*a = 0/, And\a = 0, a = -b, - 2*b + 2*a = 0/, And\a = 0, a = b, - 2*b + 2*a = 0/, And\a = -b, a = b, - 2*b + 2*a = 0/, And\a = b, b = 0, - 2*b + 2*a = 0/, And\a = -b, b = 0, - 2*b + 2*a = 0/, And\a = 0, a = -b, a = b, - 2*b + 2*a = 0/, And\a = -b, a = b, b = 0, - 2*b + 2*a = 0// < | ____ ____ | 2 / 2 2 / 2 | 2*b *\/ b 2*a *\/ a / / 2 2 \ / 2 2 \ 2 2 \ | - ------------- + ------------- for Or\And\a = 0, - 2*b + 2*a = 0/, And\b = 0, - 2*b + 2*a = 0/, - 2*b + 2*a = 0/ | 2 2 2 2 | - 3*b + 3*a - 3*b + 3*a | | 3/2 3/2 | / 2\ / 2\ | 4*\b / 4*\a / |- ----------------- + ----------------- otherwise | / 2 2\ / 2 2\ \ 3*\- 2*b + 2*a / 3*\- 2*b + 2*a /
$$\begin{cases} 0 & \text{for}\: \left(a = 0 \wedge b = 0 \wedge 2 a^{2} - 2 b^{2} = 0\right) \vee \left(a = 0 \wedge a = b \wedge b = 0 \wedge 2 a^{2} - 2 b^{2} = 0\right) \vee \left(a = 0 \wedge a = - b \wedge b = 0 \wedge 2 a^{2} - 2 b^{2} = 0\right) \vee \left(a = 0 \wedge a = - b \wedge a = b \wedge b = 0 \wedge 2 a^{2} - 2 b^{2} = 0\right) \\\frac{4 \left(a^{2}\right)^{\frac{3}{2}}}{3 \left(2 a^{2} - 2 b^{2}\right)} - \frac{4 \left(b^{2}\right)^{\frac{3}{2}}}{3 \left(2 a^{2} - 2 b^{2}\right)} & \text{for}\: \left(a = 0 \wedge a = - b\right) \vee \left(a = 0 \wedge a = b\right) \vee \left(a = 0 \wedge b = 0\right) \vee \left(a = - b \wedge a = b\right) \vee \left(a = b \wedge b = 0\right) \vee \left(a = - b \wedge b = 0\right) \vee \left(a = 0 \wedge a = - b \wedge a = b\right) \vee \left(a = 0 \wedge a = b \wedge b = 0\right) \vee \left(a = 0 \wedge a = - b \wedge b = 0\right) \vee \left(a = - b \wedge a = b \wedge b = 0\right) \vee \left(a = 0 \wedge a = - b \wedge a = b \wedge b = 0\right) \vee a = 0 \vee a = - b \vee a = b \vee b = 0 \\\sqrt{b^{2}} & \text{for}\: \left(a = - b \wedge 2 a^{2} - 2 b^{2} = 0\right) \vee \left(a = b \wedge 2 a^{2} - 2 b^{2} = 0\right) \vee \left(a = 0 \wedge a = - b \wedge 2 a^{2} - 2 b^{2} = 0\right) \vee \left(a = 0 \wedge a = b \wedge 2 a^{2} - 2 b^{2} = 0\right) \vee \left(a = - b \wedge a = b \wedge 2 a^{2} - 2 b^{2} = 0\right) \vee \left(a = b \wedge b = 0 \wedge 2 a^{2} - 2 b^{2} = 0\right) \vee \left(a = - b \wedge b = 0 \wedge 2 a^{2} - 2 b^{2} = 0\right) \vee \left(a = 0 \wedge a = - b \wedge a = b \wedge 2 a^{2} - 2 b^{2} = 0\right) \vee \left(a = - b \wedge a = b \wedge b = 0 \wedge 2 a^{2} - 2 b^{2} = 0\right) \\\frac{2 a^{2} \sqrt{a^{2}}}{3 a^{2} - 3 b^{2}} - \frac{2 b^{2} \sqrt{b^{2}}}{3 a^{2} - 3 b^{2}} & \text{for}\: \left(a = 0 \wedge 2 a^{2} - 2 b^{2} = 0\right) \vee \left(b = 0 \wedge 2 a^{2} - 2 b^{2} = 0\right) \vee 2 a^{2} - 2 b^{2} = 0 \\\frac{4 \left(a^{2}\right)^{\frac{3}{2}}}{3 \left(2 a^{2} - 2 b^{2}\right)} - \frac{4 \left(b^{2}\right)^{\frac{3}{2}}}{3 \left(2 a^{2} - 2 b^{2}\right)} & \text{otherwise} \end{cases}$$
Piecewise((0, ((a = 0)∧(b = 0)∧(-2*b^2 + 2*a^2 = 0))∨((a = 0)∧(a = b)∧(b = 0)∧(-2*b^2 + 2*a^2 = 0))∨((a = 0)∧(b = 0)∧(a = -b)∧(-2*b^2 + 2*a^2 = 0))∨((a = 0)∧(a = b)∧(b = 0)∧(a = -b)∧(-2*b^2 + 2*a^2 = 0))), (-4*(b^2)^(3/2)/(3*(-2*b^2 + 2*a^2)) + 4*(a^2)^(3/2)/(3*(-2*b^2 + 2*a^2)), (a = 0)∨(a = b)∨(b = 0)∨(a = -b)∨((a = 0)∧(a = b))∨((a = 0)∧(b = 0))∨((a = b)∧(b = 0))∨((a = 0)∧(a = -b))∨((a = b)∧(a = -b))∨((b = 0)∧(a = -b))∨((a = 0)∧(a = b)∧(b = 0))∨((a = 0)∧(a = b)∧(a = -b))∨((a = 0)∧(b = 0)∧(a = -b))∨((a = b)∧(b = 0)∧(a = -b))∨((a = 0)∧(a = b)∧(b = 0)∧(a = -b))), (sqrt(b^2), ((a = b)∧(-2*b^2 + 2*a^2 = 0))∨((a = -b)∧(-2*b^2 + 2*a^2 = 0))∨((a = 0)∧(a = b)∧(-2*b^2 + 2*a^2 = 0))∨((a = b)∧(b = 0)∧(-2*b^2 + 2*a^2 = 0))∨((a = 0)∧(a = -b)∧(-2*b^2 + 2*a^2 = 0))∨((a = b)∧(a = -b)∧(-2*b^2 + 2*a^2 = 0))∨((b = 0)∧(a = -b)∧(-2*b^2 + 2*a^2 = 0))∨((a = 0)∧(a = b)∧(a = -b)∧(-2*b^2 + 2*a^2 = 0))∨((a = b)∧(b = 0)∧(a = -b)∧(-2*b^2 + 2*a^2 = 0))), (-2*b^2*sqrt(b^2)/(-3*b^2 + 3*a^2) + 2*a^2*sqrt(a^2)/(-3*b^2 + 3*a^2), (-2*b^2 + 2*a^2 = 0)∨((a = 0)∧(-2*b^2 + 2*a^2 = 0))∨((b = 0)∧(-2*b^2 + 2*a^2 = 0))), (-4*(b^2)^(3/2)/(3*(-2*b^2 + 2*a^2)) + 4*(a^2)^(3/2)/(3*(-2*b^2 + 2*a^2)), True))
Use the examples entering the upper and lower limits of integration.