Nous allons utiliser la formule ː
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{\displaystyle \int _{a}^{b}{u'v}=[uv]_{a}^{b}-\int _{a}^{b}{uv'}}
en posant ː
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{\displaystyle u'(t)=f''(t)}
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{\displaystyle v(t)=t}
Ce qui entraîne ː
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{\displaystyle u(t)=f'(t)}
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{\displaystyle v'(t)=1}
On obtient ː
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{\displaystyle \int _{a}^{b}f''(t)\times t\,\mathrm {d} t=[f'(t)\times t]_{a}^{b}-\int _{a}^{b}f'(t)\times 1\,\mathrm {d} x}
qui s'écrit ː
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{\displaystyle \int _{a}^{b}tf''(t)\,\mathrm {d} t=bf'(b)-af'(a)-[f(t)]_{a}^{b}=bf'(b)-af'(a)-(f(b)-f(a))=\left(bf'(b)-f(b)\right)-\left(af'(a)-f(a)\right)}
![{\displaystyle \int _{a}^{b}{u'v}=[uv]_{a}^{b}-\int _{a}^{b}{uv'}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06be6d704682aaca6c244642fc6b653d938e5ead)
![{\displaystyle \int _{a}^{b}f''(t)\times t\,\mathrm {d} t=[f'(t)\times t]_{a}^{b}-\int _{a}^{b}f'(t)\times 1\,\mathrm {d} x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa46b33f77fe6f608488de1c199166d92f04214d)
![{\displaystyle \int _{a}^{b}tf''(t)\,\mathrm {d} t=bf'(b)-af'(a)-[f(t)]_{a}^{b}=bf'(b)-af'(a)-(f(b)-f(a))=\left(bf'(b)-f(b)\right)-\left(af'(a)-f(a)\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2ab2e9351ec741a6909081f75abdfc5acb278c2d)
