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int_(0)^(pi)2x cos xdx

$\int_{0}^{\pi} 2 x \cos x d x$

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Evaluate definite integrals using the chain rule

Full solution

Q.

\int_{0}^{\pi} 2 x \cos x d x

Identify integral: Step $1$ : Identify the integral to solve. $\newline$ We need to solve the integral of $2x \cos x$ from $0$ to $\pi$ . $\newline$ Calculation: $\int_{0}^{\pi} 2x \cos x \, dx$
Use integration by parts: Step $2$ : Use integration by parts. $\newline$ Let $u = 2x$ and $dv = \cos x \, dx$ . $\newline$ Then, $du = 2 \, dx$ and $v = \sin x$ . $\newline$ Calculation: $\int u \, dv = uv - \int v \, du = 2x \sin x - \int 2 \sin x \, dx$
Solve integral: Step $3$ : Solve the integral $\int 2 \sin x \, dx$ . $\newline$ Calculation: $\int 2 \sin x \, dx = -2 \cos x + C$
Substitute back: Step $4$ : Substitute back into the integration by parts formula. $\newline$ Calculation: $2x \sin x - (-2 \cos x)$ from $0$ to $\pi$ $\newline$ = $[2\pi \sin(\pi) - 2 \cos(\pi)] - [2\cdot0 \sin(0) - 2 \cos(0)]$ $\newline$ = $0 + 2 + 0 - (-2)$ $\newline$ = $4$

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