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Find \(\int x^{2} \ln (x) d(x)\)


\(u=\ln (x) \Rightarrow d u=\frac{1}{x} d(x)\)
 


\(v=\frac{x^{3}}{3} \Leftarrow d v=x^{2} d(x ) \)


\(\int u d v=u \cdot v-\int v d u \)


\(\int x^{2} \ln (x) d x=\ln (x) \frac{x^{3}}{3}-\int \frac{x^{3}}{3} \cdot \frac{1}{x} d x \)$$


\(\int {x}^{2} \ln (x) d x=\frac{1}{3} x^{3} \ln (x)-\frac{1}{3} \int x^{2} d x \)


\(\int x^{2} \ln (x) d x=\frac{1}{3} x^{3} \ln (x)-\frac{1}{3} \frac{x^{3}}{3}+C \)


\(\int x^{2} \ln (x) d x=\frac{1}{3} x^{3} \ln (x)-\frac{1}{9} x^{3}+c \)

Find \(\int x \sin x d x\)


\(u=x \quad \Longrightarrow d u=1 d x \)


\(v=-\cos x \Longleftarrow d v=\sin x d x \)


\(\int u \cdot d v=u \cdot v-\int v \cdot d u \)


\(\int x \sin x d x=x \cdot-\cos x-\int-\cos x \cdot d x \)


\(=-x \cos x+\sin x+C \)

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