- #1

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[tex]

\int \frac{x^3+6x^2+3x+16}{x^3+4x} dx

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[tex]

\int \frac{x^3+6x^2+3x+16}{x(x^2+4)} dx

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[tex]

\frac{x^3+6x^2+3x+16}{x(x^2+4)}=\frac{A}{x}+\frac{Bx+C}{x^2+4}

[/tex]

[tex]

x^3+6x^2+3x+16=A(x^2+4)+(Bx+C)x

[/tex]

[tex]

x^3+6x^2+3x+16=Ax^2+4A+Bx^2+Cx

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[tex]

A+B=6 , C=3 , A=4, B=2

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[tex]

\int \frac{4}{x}+\frac{2x+3}{x^2+4} dx

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[tex]

\int \frac{4}{x}+\frac{2x}{x^2+4} +\frac{3}{x^2+4}dx

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[tex]

4ln|x|+ln|x^2+4|+\frac{3}{2}arctan(x/2)+G

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