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

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

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Evaluate definite integrals using the chain rule

Full solution

Q.

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

Simplify using trigonometric identity: Step $1$ : Simplify the integral using a trigonometric identity. $\newline$ We know that $\cos(2\theta) = 1 - 2\sin^2(\theta)$ . However, for this integral, we directly integrate $\cos(2\theta)$ without substitution.
Integrate cosine function: Step $2$ : Integrate $\cos(2\theta)$ . $\newline$ The integral of $\cos(2\theta)$ is $\frac{1}{2}\sin(2\theta)$ plus a constant, but the constant is not needed for definite integrals.
Evaluate integral from limits: Step $3$ : Evaluate the integral from $\frac{\pi}{4}$ to $\pi$ . $\newline$ Substitute the limits into the integrated function: $\newline$ $\frac{1}{2}\sin(2\pi) - \frac{1}{2}\sin(2 \cdot \frac{\pi}{4})$ $\newline$ $= \frac{1}{2}(0) - \frac{1}{2}(0)$ $\newline$ $= 0 - 0$ $\newline$ $= 0$

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