Integración de funciones trigonométricas (3)

RESUMEN TEÓRICO
    Enunciado
  1. Calcular $\displaystyle\int \operatorname{sen}5x\cos 7x\;dx.$
  2. Calcular $\displaystyle\int \operatorname{sen}13x\operatorname{sen}8x\;dx.$
  3. Calcular $\displaystyle\int \cos (ax+b)\cos (ax-b)\;dx.$
  4. Demostrar: $$\begin{aligned}& (a)\;\operatorname{sen}px\cos qx=\dfrac{1}{2}\left[\operatorname{sen}(p+q)x+\operatorname{sen}(p-q)x\right].\\
    &(b)\;\operatorname{sen}px\operatorname{sen} qx=\dfrac{1}{2}\left[\cos(p-q)x-\cos (p+q)x\right].\\
    &(c)\;\cos px\cos qx=\dfrac{1}{2}\left[\cos (p-q)x+\cos (p+q)x\right].
    \end{aligned}$$
    Solución
  1. Usando $\operatorname{sen}px\cos qx=\frac{1}{2}\left[\operatorname{sen}(p+q)x+\operatorname{sen}(p-q)x\right]:$ $$\int \operatorname{sen}5x\cos 7x\;dx=\frac{1}{2}\int \left(\operatorname{sen}12x+\operatorname{sen}(-2x)\right)\;dx$$ $$=\frac{1}{2}\int \operatorname{sen}12x\;dx-\frac{1}{2}\int \operatorname{sen}2x\;dx=-\frac{1}{24}\cos 12x+\dfrac{1}{4}\cos 2x+C.$$
  2. Usando $\operatorname{sen}px\operatorname{sen} qx=\dfrac{1}{2}\left[\cos(p-q)x-\cos (p+q)x\right]:$ $$\int \operatorname{sen}13x\operatorname{sen}8x\;dx=\dfrac{1}{2}\int\left(\cos 5x-\cos 21x\right)\;dx$$ $$=\frac{1}{10}\operatorname{sen}5x-\frac{1}{42}\operatorname{sen}21x+C.$$
  3. Usando $\cos px\cos qx=\frac{1}{2}\left[\cos (p-q)x+\cos (p+q)x\right]:$ $$\int \cos (ax+b)\cos (ax-b)\;dx=\frac{1}{2}\int \left(\cos 2b-\cos 2ax\right)\;dx$$ $$=\frac{x\operatorname{sen}2b}{2}+\frac{\operatorname{sen}2ax}{4a}+C.$$
  4. Consideremos las conocidas fórmulas: $$\operatorname{sen}(p+q)x=\operatorname{sen}(px+qx)=\operatorname{sen}px\cos qx+\cos px\operatorname{sen}qx.\quad (1)$$ $$\operatorname{sen}(p-q)x=\operatorname{sen}(px-qx)=\operatorname{sen}px\cos qx-\cos px\operatorname{sen}qx.\quad (2)$$ $$\cos (p+q)x=\cos (px+qx)=\cos px\cos qx-\operatorname{sen}px\operatorname{sen}qx.\quad (3)$$ $$\cos (p-q)x=\cos (px-qx)=\cos px\cos qx+\operatorname{sen}px\operatorname{sen}qx.\quad (4)$$ Sumando $(1)$ y $(2):$ $\operatorname{sen}px\cos qx=\frac{1}{2}\left[\operatorname{sen}(p+q)x+\operatorname{sen}(p-q)x\right].$
    Restando $(3)$ a $(4):$ $\operatorname{sen}px\operatorname{sen} qx=\frac{1}{2}\left[\cos(p-q)x-\cos (p+q)x\right].$
    Sumando $(3)$ y $(4):$ $\cos px\cos qx=\frac{1}{2}\left[\cos (p-q)x+\cos (p+q)x\right].$
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