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(pi)/(4)int^(pi)cos(2theta)d theta

$\frac{\pi}{4} \int^{\pi} \cos (2 \theta) d \theta$

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Evaluate definite integrals using the chain rule

Full solution

Q.

\frac{\pi}{4} \int^{\pi} \cos (2 \theta) d \theta

Simplify the integral: Step $1$ : Simplify the integral. $\newline$ We need to integrate $\cos(2\theta)$ from $\frac{\pi}{4}$ to $\pi$ . $\newline$ Using the integral formula $\int \cos(ax) \, dx = \frac{1}{a}\sin(ax) + C$ , where $a = 2$ here, $\newline$ $\int \cos(2\theta) \, d\theta = \frac{1}{2}\sin(2\theta) + C$ .
Evaluate the definite integral: Step $2$ : Evaluate the definite integral. $\newline$ Plug in the limits $\frac{\pi}{4}$ and $\pi$ into $(\frac{1}{2})\sin(2\theta)$ . $\newline$ At $\theta = \pi$ , $\sin(2\pi) = 0$ . $\newline$ At $\theta = \frac{\pi}{4}$ , $\sin(\frac{\pi}{2}) = 1$ . $\newline$ So, $(\frac{1}{2})\sin(2\pi) - (\frac{1}{2})\sin(\frac{\pi}{2}) = 0 - \frac{1}{2} = -\frac{1}{2}$ .
Multiply the result: Step $3$ : Multiply the result by $\pi/4$ . Multiply $-1/2$ by $\pi/4$ to get the final answer. $(\pi/4) \times (-1/2) = -\pi/8$ .

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