First slide

Evaluation of definite integrals

Question

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

Moderate

A

$π$
B

$2 π$
C

$3 π$
D

none of these

Solution

We have,
$\begin{array}{l} 2 \sin \frac{x}{2} (\frac{1}{2} + \cos x + \cos 2 x + \dots + \cos n x) \\ = \sin \frac{x}{2} + 2 \sin \frac{x}{2} \cos x + 2 \sin \frac{x}{2} \cos 2 x + \dots + 2 \sin \frac{x}{2}, \cos n x \end{array}$
$\begin{array}{l} = \sin \frac{x}{2} + \sin \frac{3 x}{2} - \sin \frac{x}{2} + \sin \frac{5 x}{2} - \sin \frac{3 x}{2} + \dots \\ + \sin (n + \frac{1}{2}) x - \sin (n - \frac{1}{2}) x = \sin (n + \frac{1}{2}) x \end{array}$
$\frac{1}{2} + \cos x + \cos 2 x + \dots + \cos n x = \frac{\sin (n + \frac{1}{2}) x}{2 \sin (x / 2)}$
$\Rightarrow \int_{0}^{π} \frac{\sin (n + \frac{1}{2}) x}{\sin (x / 2)} d x$
$\begin{array}{l} = 2 (\int_{0}^{π} \frac{1}{2} d x + \int_{0}^{π} \cos x d x + \dots + \int_{0}^{π} \cos n x d x) \\ = 2 (\frac{π}{2} + {\sin x|}_{0}^{π} + \dots + {\frac{\sin n x}{n}|}_{0}^{π}) = π \end{array}$

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