0910&0809T3S - MAT137Y 20092010 Winter Session,...

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Unformatted text preview: MAT137Y 20092010 Winter Session, Solutions to Term Test 3 1. Evaluate the following integrals. (8%) (i) Z dx x ( ln x ) 5 / 2 . Let u = ln x . Then du = 1 x dx , thereby giving us Z du u 5 / 2 = Z u- 5 / 2 du =- 2 3 u- 3 / 2 + C =- 2 3 ( ln x )- 3 / 2 + C . (8%) (ii) Z 1 arctan xdx . We integrate by parts. Let u = arctan x , dv = dx . Then du = dx / ( 1 + x 2 ) and v = x . This yields, Z 1 arctan xdx = h x arctan x i 1- Z 1 x 1 + x 2 dx = x arctan x- 1 2 ln ( 1 + x 2 ) 1 = arctan1- 1 2 ln2- = 4- 1 2 ln2 . (10%) (iii) Z dx ( 2 + x 2 ) 2 . We apply a trig substitution: let x = 2tan . Then dx = 2sec 2 d . This gives Z 2sec 2 ( 2 + 2tan 2 ) 2 d = Z 2sec 2 4sec 4 d = 2 4 Z cos 2 d . Using trig identities, 2 4 Z cos 2 d = 2 4 1 2 + 1 4 sin2 + C = 2 8 + 2 8 sin cos + C . We substitute back in terms of x . We draw a right triangle with x being the length of the opposite side and 2 being the length of the adjacent side. By the Pythagorean theorem the length of the hypoteneuse is x 2 + 2. Therefore sin = x x 2 + 2 and cos = 2 x 2 + 2 . Hence, Z dx ( 2 + x 2 ) 2 = 2 8 arctan x 2 + 2 8 x x 2 + 2 2 x 2 + 2 + C = 2 8 arctan x 2 + 1 4 x x 2 + 2 + C . (10%) (iv) Z 1- x + 2 x 2- x 3 x ( x 2 + 1 ) 2 dx . Let I be the integral above. We re-write the integral using partial fractions: I = Z A x + Bx + C x 2 + 1 + Dx + E ( x 2 + 1 ) 2 dx . 1 We solve for the constants by grabbing a common denominator and matching coefficients with the original expression in the numerator: doing so gives us 1- x + 2 x 2- 2 x 3 = A ( x 2 + 1 ) 2 +( Bx + C ) x )( x 2 + 1 )+( Dx + E ) x = ( A + B ) x 4 + Cx 3 +( 2 A + B + D ) x 2 +( C + E ) x + A . Matching coefficients, we have A + B = , C =- 1 , 2 A + B + D = 2 , C + E =- 1 , A = 1 . Solving for the remaining constants yields B =- 1, E = 0, D = 1. Therefore I = Z 1 x- x + 1 x 2 + 1 + x ( x 2 + 1 ) 2 dx = ln | x |- 1 2 ln ( x 2 + 1 )- arctan x- 1 2 ( x 2 + 1 ) + C ....
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This note was uploaded on 04/09/2012 for the course MATH 137 taught by Professor Uppal during the Spring '08 term at University of Toronto- Toronto.

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0910&0809T3S - MAT137Y 20092010 Winter Session,...

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