3.3 Integration by parts

1.

(a) $${\frac{x^2 \log{x}}{2} - \frac{x^2}{4}}$$(b) $${-e^{x}(x^{2} + 2x + 2)}$$(c) $${x(\log{x})^2 - 2(x\log{x} - x)}$$

(d) $${\frac{1}{16}(x+5)^{16} - \frac{1}{3}(x+5)^{15}}$$(e) $${\cos (x) + x \sin (x)}$$

(f) $${-\frac{e^{x}}{2}(\cos{x} - \sin{x})}$$ (g) $${x \log{(1+x^2)} - 2(x - \tan^{-1}{x})}$$

(h) $${\frac{x^2 \tan^{-1}{x}}{2} - \frac{1}{2}(x - \tan^{-1}{x})}$$ (i) $${\frac{x^{n+1} \log{x}}{n+1} - \frac{x^{n+1}}{n+1}}$$

(j) $${-x^{3}\cos{x} + 3x^{2}\sin{x} + 6x\cos{x} - 6\sin{x}}$$ (k) $${x \cosh{x} - \sinh{x}}$$