271ms1 - MAT 271 Integral Mastery Test #1: Solutions (1)...

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(1) Transform the integral Z ln x e x dx into another integral using integration by parts. You do not need to evaluate this second integral. Solution: Let u = ln x and v 0 = e x . Then u 0 = 1 x and v = e x , so Z ln x e x dx = Z uv 0 dx = uv - Z u 0 v dx = e x ln x - Z 1 x · e x dx . (2) Peform a trigonometric substitution on the integral Z x 2 2 x + x 2 + 9 dx . You do not need to evaluate the new integral. Solution: Since we have p x 2 + 9 = p x 2 + 3 2 in the integral, we let x = 3 tan θ , so that dx/dθ = 3 sec 2 θ (and, consequently, dx = 3 sec 2 θ dθ ) and p x 2 + 9 = 3 sec θ . Substituting: Z x 2 2 x + x 2 + 9 dx = Z (3 tan θ ) 2 2(3 tan θ ) + 3 sec θ · 3 sec 2 θ dθ = Z 27 tan 2 θ sec 2 θ 6 tan θ + 3 sec θ . (3) Find the form of the Partial Fraction Decomposition of x 3 - 3 x 2 + 3 x - 1 ( x + 1) 3 ( x - 2)( x 2 - 4 x + 5) . You may keep constants such as
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271ms1 - MAT 271 Integral Mastery Test #1: Solutions (1)...

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