Evaluate: | Basic Mathematics 2081

Basic Mathematics20815 marksNumerical: integration by partsIntegration by parts

Evaluate: $\int_0^{\pi} x \sin x \, dx$ [5]

Evaluate: $$\int_0^{\pi} x \sin x , dx$$ [5]

Integrand: $x \sin x$ - Limits: from $0$ to $\pi$ Using $\int u \, dv = uv - \int v \, du$ . Let: - $u = x \Rightarrow du = dx$ - $dv = \sin x \, dx \Rightarrow v = -\cos x$ $\int0^{\pi} x \sin x \, dx = \Big[-x\cos x\Big]0^{\pi} - \int0^{\pi} (-\cos x)\, dx$ $$= \Big[-x\cos x\Big]0^{\pi} + \i...

Integrand: $x \sin x$ - Limits: from $0$ to $\pi$ Using $\int u , dv = uv - \int v , du$. Let: - $u = x \Rightarrow du = dx$ - $dv = \sin x , dx \Rightarrow v = -\cos x$ $$\int0^{\pi} x \sin x , dx = \Big[-x\cos x\Big]0^{\pi} - \int0^{\pi} (-\cos x), dx$$ $$= \Big[-x\cos x\Big]0^{\pi} + \i...