Integral de (cosaxcosbx)^2 dx
Solución
Respuesta (Indefinida)
[src]
// x for And(a = 0, b = 0)\
|| |
|| 2 2 |
|| x*cos (b*x) x*sin (b*x) cos(b*x)*sin(b*x) |
|| ----------- + ----------- + ----------------- for a = 0 |
|| 2 2 2*b |
|| |
|| 4 4 2 2 3 3 |
/ || 3*x*cos (b*x) 3*x*sin (b*x) 3*x*cos (b*x)*sin (b*x) 3*sin (b*x)*cos(b*x) 5*cos (b*x)*sin(b*x) |
| || ------------- + ------------- + ----------------------- + -------------------- + -------------------- for Or(a = -b, a = b)|
| 2 || 8 8 4 8*b 8*b |
| (cos(a*x)*cos(b*x)) dx = C + |< |
| || 2 2 |
/ || x*cos (a*x) x*sin (a*x) cos(a*x)*sin(a*x) |
|| ----------- + ----------- + ----------------- for b = 0 |
|| 2 2 2*a |
|| |
|| 3 2 3 2 3 2 3 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 2 2 2 2 2 |
||a *cos (a*x)*cos(b*x)*sin(b*x) a *sin (a*x)*cos(b*x)*sin(b*x) b *cos (b*x)*cos(a*x)*sin(a*x) b *sin (b*x)*cos(a*x)*sin(a*x) b*x*a *cos (a*x)*cos (b*x) b*x*a *cos (a*x)*sin (b*x) b*x*a *cos (b*x)*sin (a*x) b*x*a *sin (a*x)*sin (b*x) a*x*b *cos (a*x)*cos (b*x) a*x*b *cos (a*x)*sin (b*x) a*x*b *cos (b*x)*sin (a*x) a*x*b *sin (a*x)*sin (b*x) 2*a*b *cos (a*x)*cos(b*x)*sin(b*x) 2*b*a *cos (b*x)*cos(a*x)*sin(a*x) |
||------------------------------ + ------------------------------ - ------------------------------ - ------------------------------ + -------------------------- + -------------------------- + -------------------------- + -------------------------- - -------------------------- - -------------------------- - -------------------------- - -------------------------- - ---------------------------------- + ---------------------------------- otherwise |
|| 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 |
|| - 4*a*b + 4*b*a - 4*a*b + 4*b*a - 4*a*b + 4*b*a - 4*a*b + 4*b*a - 4*a*b + 4*b*a - 4*a*b + 4*b*a - 4*a*b + 4*b*a - 4*a*b + 4*b*a - 4*a*b + 4*b*a - 4*a*b + 4*b*a - 4*a*b + 4*b*a - 4*a*b + 4*b*a - 4*a*b + 4*b*a - 4*a*b + 4*b*a |
\\ /
$$\int \left(\cos{\left(a x \right)} \cos{\left(b x \right)}\right)^{2}\, dx = C + \begin{cases} x & \text{for}\: a = 0 \wedge b = 0 \\\frac{x \sin^{2}{\left(b x \right)}}{2} + \frac{x \cos^{2}{\left(b x \right)}}{2} + \frac{\sin{\left(b x \right)} \cos{\left(b x \right)}}{2 b} & \text{for}\: a = 0 \\\frac{3 x \sin^{4}{\left(b x \right)}}{8} + \frac{3 x \sin^{2}{\left(b x \right)} \cos^{2}{\left(b x \right)}}{4} + \frac{3 x \cos^{4}{\left(b x \right)}}{8} + \frac{3 \sin^{3}{\left(b x \right)} \cos{\left(b x \right)}}{8 b} + \frac{5 \sin{\left(b x \right)} \cos^{3}{\left(b x \right)}}{8 b} & \text{for}\: a = - b \vee a = b \\\frac{x \sin^{2}{\left(a x \right)}}{2} + \frac{x \cos^{2}{\left(a x \right)}}{2} + \frac{\sin{\left(a x \right)} \cos{\left(a x \right)}}{2 a} & \text{for}\: b = 0 \\\frac{a^{3} b x \sin^{2}{\left(a x \right)} \sin^{2}{\left(b x \right)}}{4 a^{3} b - 4 a b^{3}} + \frac{a^{3} b x \sin^{2}{\left(a x \right)} \cos^{2}{\left(b x \right)}}{4 a^{3} b - 4 a b^{3}} + \frac{a^{3} b x \sin^{2}{\left(b x \right)} \cos^{2}{\left(a x \right)}}{4 a^{3} b - 4 a b^{3}} + \frac{a^{3} b x \cos^{2}{\left(a x \right)} \cos^{2}{\left(b x \right)}}{4 a^{3} b - 4 a b^{3}} + \frac{a^{3} \sin^{2}{\left(a x \right)} \sin{\left(b x \right)} \cos{\left(b x \right)}}{4 a^{3} b - 4 a b^{3}} + \frac{a^{3} \sin{\left(b x \right)} \cos^{2}{\left(a x \right)} \cos{\left(b x \right)}}{4 a^{3} b - 4 a b^{3}} + \frac{2 a^{2} b \sin{\left(a x \right)} \cos{\left(a x \right)} \cos^{2}{\left(b x \right)}}{4 a^{3} b - 4 a b^{3}} - \frac{a b^{3} x \sin^{2}{\left(a x \right)} \sin^{2}{\left(b x \right)}}{4 a^{3} b - 4 a b^{3}} - \frac{a b^{3} x \sin^{2}{\left(a x \right)} \cos^{2}{\left(b x \right)}}{4 a^{3} b - 4 a b^{3}} - \frac{a b^{3} x \sin^{2}{\left(b x \right)} \cos^{2}{\left(a x \right)}}{4 a^{3} b - 4 a b^{3}} - \frac{a b^{3} x \cos^{2}{\left(a x \right)} \cos^{2}{\left(b x \right)}}{4 a^{3} b - 4 a b^{3}} - \frac{2 a b^{2} \sin{\left(b x \right)} \cos^{2}{\left(a x \right)} \cos{\left(b x \right)}}{4 a^{3} b - 4 a b^{3}} - \frac{b^{3} \sin{\left(a x \right)} \sin^{2}{\left(b x \right)} \cos{\left(a x \right)}}{4 a^{3} b - 4 a b^{3}} - \frac{b^{3} \sin{\left(a x \right)} \cos{\left(a x \right)} \cos^{2}{\left(b x \right)}}{4 a^{3} b - 4 a b^{3}} & \text{otherwise} \end{cases}$$
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.