Lecture3-slides-IntegrationByParts - Warm-Up Problem Find the following indefinite integeral \u221a Z x \u221a dx x 7 Announcement Homework 1 is due on

# Lecture3-slides-IntegrationByParts - Warm-Up Problem...

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Warm-Up Problem Find the following indefinite integeral. Z x x + 7 dx Announcement Homework 1 is due on Thursday, at 8:45 am. Homework must be submitted on Gradescope.
Lecture 3 I Integration By Parts ( § 6.1) I Trigonometric Integrals ( § 6.2) I Trigonometric Substitution ( § 6.2) (if time permits)
Integration by Parts I Substitution “the reverse of chain rule” I Integration by parts “the reverse of product rule”
Integration by Parts I Substitution “the reverse of chain rule” I Integration by parts “the reverse of product rule” Motivation/Examples 1. Z xe x dx = ? 2. Z w cos( w ) dw = ? 3. Z e t sin( t ) dt = ?
Recall: Product Rule d dx [ f ( x ) g ( x )] =
Recall: Product Rule d dx [ f ( x ) g ( x )] = f 0 ( x ) g ( x ) + f ( x ) g 0 ( x )
Recall: Product Rule d dx [ f ( x ) g ( x )] = f 0 ( x ) g ( x ) + f ( x ) g 0 ( x ) Take the integral of both sides: Z d dx [ f ( x ) g ( x )] dx = Z f 0 ( x ) g ( x ) + f ( x ) g 0 ( x ) dx
Recall: Product Rule d dx [ f ( x ) g ( x )] = f 0 ( x ) g ( x ) + f ( x ) g 0 ( x ) Take the integral of both sides: Z d dx [ f ( x ) g ( x )] dx = Z f 0 ( x ) g ( x ) + f ( x ) g 0 ( x ) dx f ( x ) g ( x )
Recall: Product Rule d dx [ f ( x ) g ( x )] = f 0 ( x ) g ( x ) + f ( x ) g 0 ( x ) Take the integral of both sides: Z d dx [ f ( x ) g ( x )] dx = Z f 0 ( x ) g ( x ) + f ( x ) g 0 ( x ) dx f ( x ) g ( x ) = Z f 0 ( x ) g ( x ) dx + Z f ( x ) g 0 ( x ) dx
Recall: Product Rule d dx [ f ( x ) g ( x )] = f 0 ( x ) g ( x ) + f ( x ) g 0 ( x ) Take the integral of both sides: Z d dx [ f ( x ) g ( x )] dx = Z f 0 ( x ) g ( x ) + f ( x ) g 0 ( x ) dx f ( x ) g ( x ) = Z f 0 ( x ) g ( x ) dx + Z f ( x ) g 0 ( x ) dx
Recall: Product Rule d dx [ f ( x ) g ( x )] = f 0 ( x ) g ( x ) + f ( x ) g 0 ( x ) Take the integral of both sides: Z d dx [ f ( x ) g ( x )] dx = Z f 0 ( x ) g ( x ) + f ( x ) g 0 ( x ) dx f ( x ) g ( x ) = Z f 0 ( x ) g ( x ) dx + Z f ( x )