Techniques of Integration - Integration by Parts

Thomas J. Pfaff ()
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Thomas J. Pfaff: Ithaca College, Department of Mathematics

Chapter Chapter 32 in Applied Calculus with R, 2023, pp 363-366 from Springer

Abstract: Abstract In deriving a formula for reversing the product rule we start with the product rule $$ (f(x)g(x))' =f'(x)g(x) + f(x)g'(x) $$ ( f ( x ) g ( x ) ) ′ = f ′ ( x ) g ( x ) + f ( x ) g ′ ( x ) and then integrate both sides with respect to x to get $$ \int (f(x)g(x))'dx =\int f'(x)g(x) dx + \int f(x)g'(x) dx. $$ ∫ ( f ( x ) g ( x ) ) ′ d x = ∫ f ′ ( x ) g ( x ) d x + ∫ f ( x ) g ′ ( x ) d x . Now $$\int (f(x)g(x))'dx =f(x)g(x)$$ ∫ ( f ( x ) g ( x ) ) ′ d x = f ( x ) g ( x ) and so $$ f(x)g(x) =\int f'(x)g(x) dx + \int f(x)g'(x) dx $$ f ( x ) g ( x ) = ∫ f ′ ( x ) g ( x ) d x + ∫ f ( x ) g ′ ( x ) d x or $$ \int f(x)g'(x) dx = f(x)g(x) - \int f'(x)g(x) dx. $$ ∫ f ( x ) g ′ ( x ) d x = f ( x ) g ( x ) - ∫ f ′ ( x ) g ( x ) d x .

Date: 2023
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DOI: 10.1007/978-3-031-28571-4_32

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