Calculus With Analytic Geometry II Quiz #3 Solutions PDF

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Quiz #3 Solutions...

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Calculus With Analytic Geometry II Quiz #3–Solutions

INSTRUCTIONS. • Show as much work as possible. No work could mean no credit. This is worth repeating:

WORK!

SHOW

• Write clearly, preferably in full sentences. Disconnected expressions could cost you points; use the equal sign where appropriate. • Improper use of the equal sign will cost you points. • Avoid: Double columns, arrows. If you need additional paper, ask for it. Compute the following integrals (25 points each): ∫ 1. x sin 2x dx. Solution. ∫ 2.

∫

1 1 x sin 2x dx = − x cos 2x + 2 2

∫

1 1 cos 2x dx = − x cos 2x + sin 2x + C . 4 2

x2 ln x dx.

Solution. ∫ 3.

∫

1 1 x2 ln x dx = x2 ln x − 3 3

∫

x3

1 1 1 dx = x3 ln x − x3 + C. 9 3 x

ln 2

xex dx.

0

Solution. ∫ 4.

∫

ln 2 0

 ln 2 ∫ xex dx = xex − 0

0

ln 2

 ln 2 ex dx = 2 ln 2 − ex = 2 ln 2 − (2 − 1) = 2 ln 2 − 1. 0

sin2 θ cos5 θ dθ .

Solution. ∫

Substitute u = sin θ, du = cos θ dθ. Then ∫ ∫ ∫ sin2 θ cos5 θ dθ = sin2 θ (1 − sin2 θ )2 cos θ dθ = u2 (1 − u2 )2 du = (u2 − 2u4 + u6 ) du = =

1 3 2 5 1 7 u − u + u +C 7 5 3 1 2 1 3 sin θ − sin5 θ + sin7 θ + C. 5 3 7...