Integrate the function: x sin x | vedclass

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Question

(A) Let $I = \int x \sin x \, dx$. Using the integration by parts formula: $\int u v \, dx = u \int v \, dx - \int \left( \frac{du}{dx} \int v \, dx \

Vedclass · Accepted Answer

$-x \cos x + \sin x + C$

Vedclass · Answer

(A) Let $I = \int x \sin x \, dx$. Using the integration by parts formula: $\int u v \, dx = u \int v \, dx - \int \left( \frac{du}{dx} \int v \, dx \right) dx$. Taking $u = x$ (first function) and $v = \sin x$ (second function),we get: $I = x \int \sin x \, dx - \int \left( \frac{d}{dx}(x) \int \sin x \, dx \right) dx$ $I = x(-\cos x) - \int (1 \cdot (-\cos x)) \, dx$ $I = -x \cos x + \int \cos x \, dx$ $I = -x \cos x + \sin x + C$,where $C$ is an arbitrary constant.

Integrate the function: $x \sin x$

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