 Sec. 4.5: Integration by Substitution. T HEOREM 4.12 Antidifferentiation of a Composite Function Let g be a function whose range is an interval I, and.

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Sec. 4.5: Integration by Substitution

T HEOREM 4.12 Antidifferentiation of a Composite Function Let g be a function whose range is an interval I, and let f be a function that is continuous on I. If g is differentiable on its domain and F is an antiderivative of f on I, then

Sec. 4.5: Integration by Substitution

Rewrite the integral:

Sec. 4.5: Integration by Substitution When solving a differential equation of the form it’s convenient to write it of the form

Sec. 4.5: Integration by Substitution Find the differential dy.

Sec. 4.5: Integration by Substitution u = 7x du = 7 dx

Sec. 4.5: Integration by Substitution u = x 2 – 4 du = 2x dx 1/2 du = x dx

Sec. 4.5: Integration by Substitution u = sin 3x du = (cos 3x) 3 dx 1/3 du = cos 3x dx

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Sec. 4.5: Integration by Substitution u = 2x – 1 du = 2 dx u + 1 = 2x 1/2 du = dx (u + 1) / 2 = x

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