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DODGES, PITFALLS, AND TRIUMPHS

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Examples.

(1) Find $\int w\cdot \sin w\,dw$ .

Write $u=w$ , and for $\sin w\cdot dw$ write $dx$ . We shall then have du=dw, while $\int \sin w\cdot dw=-\cos w=x$ .

Putting these into the formula, we get

${\begin{aligned}\int w\cdot \sin w\,dw&=w(-\cos w)-\int -\cos w\,dw\\&=-w\cos w+\sin w+C.\end{aligned}}$

(2) Find $\int x\epsilon ^{x}\,dx$ .

${\begin{aligned}{\text{Write}}\;\quad \quad \quad u&=x,&\epsilon ^{x}\,dx&=dv;\\{\text{then }}\;\quad \quad \quad du&=dx,&v&=\epsilon ^{x},\end{aligned}}$

and

${\begin{aligned}\int x\epsilon ^{x}\,dx=x\epsilon ^{x}-\int \epsilon ^{x}\,dx\quad {\text{(by the formula)}}\\=x\epsilon ^{x}-\epsilon ^{x}=\epsilon ^{x}(x-1)+C.\end{aligned}}$

(3) Try $\int \cos ^{2}\theta \,d\theta$ .

${\begin{aligned}u&=\cos \theta ,\cos \theta \,\quad \quad d\theta =dv.\\{\text{Hence }}\;\quad \quad \quad du&=-\sin \theta \,d\theta ,\quad v=\sin \theta ,\\\int \cos ^{2}\theta \,d\theta &=\cos \theta \sin \theta +\int \sin ^{2}\theta \,d\theta \\&={\frac {2\cos \theta \sin \theta }{2}}+\int (1-\cos ^{2}\theta )\,d\theta \\&={\frac {\sin 2\theta }{2}}+\int d\theta -\int \cos ^{2}\theta \,d\theta .\end{aligned}}$

${\begin{aligned}{\text{Hence}}\;\quad \quad \quad 2\int \cos ^{2}\theta \,d\theta &={\frac {\sin 2\theta }{2}}+\theta \\and\quad \quad \quad \quad \int \cos ^{2}\theta \,d\theta &={\frac {\sin 2\theta }{4}}+{\frac {\theta }{2}}+C.\end{aligned}}$

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