8.4-PFD-part2

8.4-PFD-part2 - MthSc 108-Learning Activity (Part 2)...

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MthSc 108-Learning Activity (Part 2) Section 8.4: Integration of Rational Functions by Partial Fractions 1. Evaluate dx x + 1 ( ) x 2 + 4 ( ) . ( )( ) ( ) ( )( ) ( ) ( ) ( )( ) ( )( ) 2 2 2 2 2 2 2 2 1 1 4 1 4 1 4 1 4 0 4 1 1 1 1 0 5 1 , 5 5 5 0 4 1 1 1 1 1 1 1 1 1 5 5 5 1 5 1 5 4 4 1 4 1 1 1 1 5 1 5 1 4 A Bx C x x x x A x Bx C x A B x B C x A C A B B C B C B B A C B C A C x x x x x x x x dx x x x x + = + + + + + = + + + + = + + + + + + = - + = + = - = = - = = + = + = - + - = + = + + + + + + + - = + + + + ( ) 2 2 2 2 2 4 1 1 1 1 1 5 1 5 5 4 4 1 1 1 1 1 5 1 5 5 4 4 1 2 1 Let in the second integral. Then dx dx=2d 2 2 dx x x dx dx dx x x x x dx dx x x x x u du + = + - + + + = + - + + + = = ( )( ) ( )( ) ( ) 2 1 2 2 1 2 2 u. 1 Also, let +4 in the third integral. Then 2 , and 2 1 1 1 1 1 1 1 = ln 1 ln 1 tan ln 5 10 10 5 10 10 1 1 4 1 1 1 ln 1 tan ln 4 2 5 10 10 1 4 w x dw xdx dw xdx dx dw x du x u w C w u x x dx x x x C x x - - = = = + + - = + + - + + + + = + + - + + + +
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2. Evaluate x 2 dx x 2 + 4 ( ) 2 .
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8.4-PFD-part2 - MthSc 108-Learning Activity (Part 2)...

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