# MATH 16B Lecture Notes - Lecture 14: Scilab, Antiderivative, Product Rule

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○ Integration by parts = product rule in reverse

○ Product Rule: d/dx (f(x)g(x)) = f(x)g'(x) + f'(x)g(x)

■ Take antiderivatives of both sides:

■ f(x)g(x) = integral of f(x)g'(x)dx + integral of f'(x)g(x)dx

■ Integral of f(x)g'(x)dx = f(x)g(x) - integral of f'(x)g(x)dx

■ Alternate Form: G'(x) = g(x) where G(x) is the antiderivative of g(x)

■ Integral of f(x)g(x)dx = f(x)G(x) - integral of f'(x)G(x)dx

■ Integration by parts formula

○ When to use:

■ Integrating a product f(x)g(x) such that we can easily find G(x), an antiderivative

of g(x)

■ Integral of f'(x)G(x)dx is easier to calculate

○ Examples:

■ Integral of xcos(x)dx = ?

■ f(x) = x and g(x) = cos(x)

■ f'(x) = 1 and G(x) = sin(x)

■ = xsin(x) - integral of 1 * sin(x)dx

■ = xsin(x) - (-cos(x)) + C

■ =xsin(x) + cos(x) + C

■ Assigning the opposite functions for f(x) and g(x) results in a more

difficult equation

■ Integral of xe^2xdx = ?

■ f(x) = x and g(x) = e^2x

■ f'(x) = 1 and G(x) = 1/2e^2x

■ = 1/2xe^2x - integral of 1/2e^2xdx

■ = 1/2xe^2x - 1/4e^2x + C

○ Sometimes it is not at all clear what f(x) and g(x) to choose

■ Integral of ln(x)dx = ?

■ f(x) = ln(x) and g(x) = 1

■ f'(x) = 1/x and G(x) = x

■ = xln(x) - integral of 1/x * x dx

■ = xln(x) - x + C

■ Integral of xln(x)dx = ?

■ f(x) = ln(x) and g(x) = x

■ f'(x) = 1/x and G(x) = 1/2x^2

■ =1/2x^2ln(x) - integral of 1/2xdx

■ = 1/2x^2ln(x) - 1/4x^2 + C

○ Sometimes it may be necessary to do integration by parts more than once

■ Integral of x^2e^xdx

■ f(x) = x^2 and g(x) = e^x

■ f'(x) = 2x and G(x) = e^x

■ = x^2e^x - integral of 2xe^xdx

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