Let α∈$$0$$,π$$2$$ be fixed. If ∫$$sin$$⁡(x+$$α$$)$$sin$$⁡(x$$−α$$)dx=f1(x)$$cos$$⁡$$2$$α$$+f2(x)$$$$sin$$⁡$$2$$α$$+c$$ where c constant of integration then

Question

Let α∈$$0$$,π$$2$$ be fixed. If ∫$$sin$$⁡(x+$$α$$)$$sin$$⁡(x$$−α$$)dx=f1(x)$$cos$$⁡$$2$$α$$+f2(x)$$$$sin$$⁡$$2$$α$$+c$$ where c constant of integration then

Infinity Learn · Accepted Answer

∫$$sin$$⁡(x+$$α$$)$$sin$$⁡(x$$−α$$)dx=$$∫$$sin$$⁡$$(x$$−α$$+2$$α$$)$$sin$$⁡(x$$−α$$)dx$$ Put x−α$$=$$θ⇒$$dx=d$$θ$$=$$∫$$sin$$⁡$$($$θ$$+2$$α$$)$$sin$$⁡θdθ $$=$$$$cos$$⁡$$2$$α∫dθ$$+$$$$sin$$⁡$$2$$α∫$$cos$$⁡θ$$sin$$⁡θdθ$$=$$$$cos$$⁡$$2$$α⋅(x$$−α$$)+$$$$sin$$⁡$$2$$αlog⁡|$$sin$$⁡$$(x$$−α$$)|$$+c$$

$Let α \in (0, \frac{π}{2}) be fixed. If \int \frac{\sin (x + α)}{\sin (x - α)} d x = f_{1} (x) \cos 2 α + f_{2} (x) \sin 2 α + c where c constant of integration then$

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Let α∈0,π2 be fixed. If ∫sin⁡(x+α)sin⁡(x−α)dx=f1(x)cos⁡2α+f2(x)sin⁡2α+c where c constant of integration then

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$Let α \in (0, \frac{π}{2}) be fixed. If \int \frac{\sin (x + α)}{\sin (x - α)} d x = f_{1} (x) \cos 2 α + f_{2} (x) \sin 2 α + c where c constant of integration then$