- #1

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**1. ∫[itex]\frac{dx}{x**

^{3}+ 2x}[/itex]We're suppose to evaluate the integral.

Use Partial Fraction Decomposition:

[itex]\frac{1}{x

^{3}+ 2x}[/itex] = [itex]\frac{A}{x}[/itex] + [itex]\frac{Bx + C}{x

^{2}+ 2}[/itex]

1 = A(x

^{2}+ 2) + (Bx + C)(x)

1 = Ax

^{2}+ 2A + Bx

^{2}+ Cx

1 = x

^{2}( A + B) + Cx + 2A

Solving for A gives [itex]\frac{1}{2}[/itex]

Solving for B gives -[itex]\frac{1}{2}[/itex]

Solving for C gives 0

∫[itex]\frac{dx}{x(x

^{2}+ 2}[/itex] = [itex]\frac{1}{2}[/itex]∫[itex]\frac{dx}{x}[/itex] - [itex]\frac{1}{2}[/itex]∫[itex]\frac{dx}{x

^{2}+ 2}[/itex]

When we evaluate this, I get:

[itex]\frac{1}{2}[/itex]ln x - [itex]\frac{1}{2}[/itex]tan

^{-1}[itex]\frac{x}{\sqrt{2}}[/itex]

Or should it be:

[itex]\frac{1}{2}[/itex]ln x - [itex]\frac{1}{2}[/itex]ln (x

^{2}+ 2)