10th Edition

Ron Larson + 1 other

ISBN: 9781285057095

Textbook Problem

Proof Suppose that f ( a ) = f ( b ) = g ( a ) = g ( b ) = 0 and the second derivatives of f and g are continuous on the closed interval [ a , b ] . Prove that

∫ a b f ( x ) g n ( x ) d x = ∫ a b f ″ ( x ) g ( x ) d x .

To determine

To prove: ∫abf(x)g′′(x)dx=∫abf′′(x)g(x)dx.

The functions f(a)=f(b)=g(a)=g(b)=0 and the second derivatives of f and g are continuous on the interval [a,b].

Formula used:

The integration by parts:

∫u⋅v=u∫v−∫(∫v)u′

Take the left-hand side of the relation.

∫abf(x)g′′(x)dx

Here u=f(x),v=g′′(x).

Use integration by parts.

∫abf(x)g′′(x)dx=[(f(x))∫g′′(x)dx]ab−∫ab{(∫g′′(x)dx)⋅ddxf(x)}dx=[f(x)g(x)]ab−∫ab{g′(x)⋅f′(x)}dx=f(b)g(b)−f(a)g(a)−∫abf′(x)⋅g′(x)dx=−∫abf′(x)⋅g′(x)dx

Again, use integration by parts

Check out a sample textbook solution.

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Proof Suppose that f ( a ) = f ( b ) = g ( a ) = g ( b ) = 0 and the second derivatives of f and g are continuous on the closed interval [ a , b ] . Prove that

∫ a b f ( x ) g n ( x ) d x = ∫ a b f ″ ( x ) g ( x ) d x .

Chapter 8 Solutions

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Proof Suppose that f ( a ) = f ( b ) = g ( a ) = g ( b ) = 0 and the second derivatives of f and g are continuous on the closed interval [ a , b ] . Prove that ∫ a b f ( x ) g n ( x ) d x = ∫ a b f ″ ( x ) g ( x ) d x .

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Calculus Calculus 10th Edition

Proof Suppose that f ( a ) = f ( b ) = g ( a ) = g ( b ) = 0 and the second derivatives of f and g are continuous on the closed interval [ a , b ] . Prove that

∫ a b f ( x ) g n ( x ) d x = ∫ a b f ″ ( x ) g ( x ) d x .