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271ms2

# 271ms2 - MAT 271 Integral Mastery Test#2 Solutions 1(1...

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MAT 271 Integral Mastery Test #2: Solutions (1) Rewrite the integral Z 1 0 x 2 sin - 1 xdx in terms of another integral using integration by parts. You do not need to evaluate this second integral. Solution: Use integration by parts, with u = sin - 1 x and v 0 = x 2 . Then u 0 = 1 1 - x 2 and v = x 3 3 and Z 1 0 x 2 sin - 1 xdx = x 3 3 · sin - 1 x ± ± ± ± 1 0 - Z 1 0 x 3 3 · 1 1 - x 2 dx (2) Peform a trigonometric substitution on the integral Z y y 2 - p y 2 - 16 dy . You do not need to evaluate the new integral. Solution: The trigonometric substitution you need to use is y = 4sec θ . Then dy = 4sec θ tan θ dθ and p y 2 - 16 = 4tan θ , and Z y y 2 - p y 2 - 16 dy = Z 4sec θ 16sec 2 θ - 4tan θ · 4sec θ tan θ dθ = Z 16sec 2 θ tan θ 16sec 2 θ - 4tan θ 1

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(3) Find the form of the Partial Fraction Decomposition of 2 x 3 + 5 x - 7 x ( x - 4) 2 ( x 2 - 2 x + 7) 2 . You may keep constants such as A , B , C , ... in your answer. (Note that x 2 - 2 x + 7 has no real roots.) Solution: A x + B x - 4 + C ( x - 4) 2 + Dx + E x 2 - 2 x + 7 + Fx + G ( x 2 - 2 x + 7) 2 (4) Rewrite the integral
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271ms2 - MAT 271 Integral Mastery Test#2 Solutions 1(1...

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