Mathematics

Evaluate: $$\int x \sin 2 x d x$$


SOLUTION
$$I=\int x\sin 2x$$

We know the formula (by Parts)
$$\displaystyle\int uv=u\int v-\int \left(\int v\right)\dfrac{du}{dn}dx$$

$$u=x;v\sin 2x$$

$$I=x\displaystyle \int \sin 2x-\int \left(\int \sin 2x\right)\dfrac{d}{dx}(x)dx$$

$$I=x\dfrac{(-\cos (2x))}{2}-\displaystyle\int \left(\dfrac{-\cos 2x}{2}\right)1dn$$

$$I=\dfrac{-x\cos 2x}{2}+\dfrac{\sin 2x}{2.2}+C$$

$$I=\dfrac{\sin 2x}{4}-\dfrac{x\cos 2x}{2}+C$$.
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Subjective Medium Published on 17th 09, 2020
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