(b) Use integration by parts to evaluate integrate x * cos x dx from 0 to pi/2

Set up integration formula: Step 1: Set up the integration by parts formula. We use the formula ∫u dv = uv - ∫v du. Let u = x and dv = cos(x) dx. Then, du = dx and v = ∫cos(x) dx = sin(x).Apply integration by parts: Step 2: Apply the integration by parts formula. Plug in u, v, du, and dv into the formula: ∫x cos(x) dx = x sin(x) - ∫sin(x) dx.Integrate sin(x): Step 3: Integrate ∫sin(x) dx. The integral of sin(x) is -cos(x), so: ∫x cos(x) dx = x sin(x) + cos(x).Evaluate definite integral: Step 4: Evaluate the definite integral from 0 to π/2. Plug in the limits of integration: [x sin(x) + cos(x)] from 0 to π/2. = (π/2 * sin(π/2) + cos(π/2)) - (0 * sin(0) + cos(0)) = (π/2 * 1 + 0) - (0 * 0 + 1) = π/2 - 1.

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Find indefinite integrals using the substitution and by parts

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Q. (b) Use integration by parts to evaluate

\int_0^{\frac{\pi}{2}} x \cdot \cos x \, dx

Set up integration formula: Step $1$ : Set up the integration by parts formula. $\newline$ We use the formula $\int u \, dv = uv - \int v \, du$ . Let $u = x$ and $dv = \cos(x) \, dx$ . $\newline$ Then, $du = dx$ and $v = \int\cos(x) \, dx = \sin(x)$ .
Apply integration by parts: Step $2$ : Apply the integration by parts formula. $\newline$ Plug in $u$ , $v$ , $du$ , and $dv$ into the formula: $\newline$ $\int x \cos(x) \, dx = x \sin(x) - \int \sin(x) \, dx$ .
Integrate $\sin(x)$ : Step $3$ : Integrate $\int \sin(x) \, dx$ . The integral of $\sin(x)$ is $-\cos(x)$ , so: $\int x \cos(x) \, dx = x \sin(x) + \cos(x)$ .
Evaluate definite integral: Step $4$ : Evaluate the definite integral from $0$ to $\pi/2$ . $\newline$ Plug in the limits of integration: $\newline$ $[x \sin(x) + \cos(x)]$ from $0$ to $\pi/2$ . $\newline$ = $(\pi/2 \cdot \sin(\pi/2) + \cos(\pi/2)) - (0 \cdot \sin(0) + \cos(0))$ $\newline$ = $(\pi/2 \cdot 1 + 0) - (0 \cdot 0 + 1)$ $\newline$ = $\pi/2 - 1$ .