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

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21 Apr 2017
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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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