29.10. Compute using integration by parts:

a) $\int x^{3} \ln x d x$; b) $\int \operatorname{arctg} x d x$; c) $\int x \cos x d x$; d) $\int x e^{x} d x$.
29.10. а) $\int x^{3} \ln x d x=\int \ln x d\left(\frac{x^{4}}{4}\right)=\ln x \cdot \frac{x^{4}}{4}-\int \frac{x^{4}}{4} d(\ln x)=$ $=\ln x \cdot \frac{x^{4}}{4}-\int \frac{x^{4}}{4} \cdot \frac{1}{x} d x=\frac{x^{4}}{4} \ln x-\frac{x^{4}}{16}+C$.

б) $\int \operatorname{arctg} x d x=x \operatorname{arctg} x-\int \frac{x d x}{1+x^{2}}=x \operatorname{arctg} x-\frac{1}{2} \ln \left(1+x^{2}\right)+C$.

в) $\int x \cos x d x=\int x d(\sin x)=x \sin x-\int \sin x d x=x \sin x+\cos x+C$.

г) $\int x e^{x} d x=\int x d\left(e^{x}\right)=x e^{x}-\int e^{x} d x=x e^{x}-e^{x}+C$.

29.10. a) $\int x^{3} \ln x d x=\int \ln x d\left(\frac{x^{4}}{4}\right)=\ln x \cdot \frac{x^{4}}{4}-\int \frac{x^{4}}{4} d(\ln x)=$ $=\ln x \cdot \frac{x^{4}}{4}-\int \frac{x^{4}}{4} \cdot \frac{1}{x} d x=\frac{x^{4}}{4} \ln x-\frac{x^{4}}{16}+C$.

b) $\int \operatorname{arctg} x d x=x \operatorname{arctg} x-\int \frac{x d x}{1+x^{2}}=x \operatorname{arctg} x-\frac{1}{2} \ln \left(1+x^{2}\right)+C$.

c) $\int x \cos x d x=\int x d(\sin x)=x \sin x-\int \sin x d x=x \sin x+\cos x+C$.

d) $\int x e^{x} d x=\int x d\left(e^{x}\right)=x e^{x}-\int e^{x} d x=x e^{x}-e^{x}+C$.