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week11 - More on Integration 1 Simplify the integrand...

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More on Integration 1. Simplify the integrand before integrating. ex. Z 9 x 2 + 5 3 x dx = Z ± 3 x + 5 3 x - 1 ² dx = 3 x 2 2 + 5 3 ln | x | + C ex . Z 6 ( e 4 - 3 x ) 2 dx = Z 6 e 2(4 - 3 x ) dx = Z 6 e 8 - 6 x dx = Z 6 e u du - 6 , where u = 8 - 6 x ; du = - 6 dx ; dx = du - 6 = - 6 6 Z e u du = - e u + C = - e 2(4 - 3 x ) + C or = - ( e 4 - 3 x ) 2 + C ex . Z x 4 x 2 + 1 dx = Z x ( x 2 + 1) - 1 / 4 dx = Z u - 1 / 4 du 2 , where u = x 2 + 1; du = 2 xdx ; xdx = du 2 = 1 2 u - 1 / 4+1 3 / 4 + C = 2 3 ( x 2 + 1) 3 / 4 + C 2. Integrals involving ln x ex. Z 4 x ln(2 x 2 ) dx = 4 Z 1 x ln(2 x 2 ) dx = I Let u = ln(2 x 2 ). du = 1 2 x 2 · d dx (2 x 2 ) = 1 2 x 2 (4 x ) dx = 2 x dx . Thus, 1 x dx = du 2 I = 4 Z 1 u · du 2 = 4 2 Z 1 u du = 2 ln | u | + C = 2 ln | ln(2 x 2 ) | + C

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3. Integrals involving b v ex . Z b v dv = Z ( e ln b ) v dv, since b = e ln b = Z e v ln b dv = Z e u du ln b , where u = v ln b ; du = (ln b ) dv, thus dv = du ln b = 1 ln b Z e u du, since (ln b ) is a constant = 1 ln b e u + C = 1
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week11 - More on Integration 1 Simplify the integrand...

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