Examples.
(1) Find ∫ w ⋅ sin w d w {\displaystyle \int w\cdot \sin w\,dw} .
Write u = w {\displaystyle u=w} , and for sin w ⋅ d w {\displaystyle \sin w\cdot dw} write d x {\displaystyle dx} . We shall then have du=dw, while ∫ sin w ⋅ d w = − cos w = x {\displaystyle \int \sin w\cdot dw=-\cos w=x} .
Putting these into the formula, we get
∫ w ⋅ sin w d w = w ( − cos w ) − ∫ − cos w d w = − w cos w + sin w + C . {\displaystyle {\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 ∫ x ϵ x d x {\displaystyle \int x\epsilon ^{x}\,dx} .
Write u = x , ϵ x d x = d v ; then d u = d x , v = ϵ x , {\displaystyle {\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
∫ x ϵ x d x = x ϵ x − ∫ ϵ x d x (by the formula) = x ϵ x − ϵ x = ϵ x ( x − 1 ) + C . {\displaystyle {\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 ∫ cos 2 θ d θ {\displaystyle \int \cos ^{2}\theta \,d\theta } .
u = cos θ , cos θ d θ = d v . Hence d u = − sin θ d θ , v = sin θ , ∫ cos 2 θ d θ = cos θ sin θ + ∫ sin 2 θ d θ = 2 cos θ sin θ 2 + ∫ ( 1 − cos 2 θ ) d θ = sin 2 θ 2 + ∫ d θ − ∫ cos 2 θ d θ . {\displaystyle {\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}}}
Hence 2 ∫ cos 2 θ d θ = sin 2 θ 2 + θ a n d ∫ cos 2 θ d θ = sin 2 θ 4 + θ 2 + C . {\displaystyle {\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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, and for 
write 
. We shall then have du=dw, while 
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