The value of ∫$$0$$π $$sin$$⁡(n+1/2)xsin$$⁡$$(x/2)dx(n$$∈$$N) is

Question

The value of ∫$$0$$π $$sin$$⁡(n+1/2)xsin$$⁡$$(x/2)dx(n$$∈$$N) is

Infinity Learn · Accepted Answer

We have,$$2sin$$⁡$$x212+$$$$cos$$⁡$$x+$$$$cos$$⁡$$2x+$$…$$+$$$$cos$$⁡$$nx=$$$$sin$$⁡$$x2+2sin$$⁡$$x2cos$$⁡$$x+2sin$$⁡$$x2cos$$⁡$$2x+$$…$$+2sin$$⁡$$x2$$,$$cos$$ $$nx=$$$$sin$$⁡$$x2+$$$$sin$$⁡$$3x2$$−$$sin$$⁡$$x2+$$$$sin$$⁡$$5x2$$−$$sin$$⁡$$3x2+$$…$$+$$$$sin$$⁡$$n+12x$$−$$sin$$⁡n−$$12x=$$$$sin$$⁡$$n+12x12+$$$$cos$$⁡$$x+$$$$cos$$⁡$$2x+$$…$$+$$$$cos$$⁡$$nx=$$$$sin$$⁡$$n+12x2sin$$⁡(x/2)⇒ ∫$$0$$π $$sin$$⁡$$n+12xsin$$⁡$$(x/2)dx=2$$∫$$0$$π $$12dx+$$∫$$0$$π $$cos$$⁡$$xdx+$$…$$+$$∫$$0$$π $$cos$$⁡$$nxdx=2$$$$π$$$$2+$$$$sin$$⁡$$x0$$π$$+$$…$$+$$$$sin$$⁡$$nxn0$$π$$=$$π

The value of $\int_{0}^{π} \frac{\sin (n + 1 / 2) x}{\sin (x / 2)} d x (n \in N)$ is

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The value of ∫0π sin⁡(n+1/2)xsin⁡(x/2)dx(n∈N) is

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The value of $\int_{0}^{π} \frac{\sin (n + 1 / 2) x}{\sin (x / 2)} d x (n \in N)$ is