Use the method of partial fractions to verify the following integrals. Hint: First perform polynomial long division on the integrand, then factor the denominator where appropriate (show all steps). \int 4x ^2 + 2x − 1 / x ^3 + x ^2 dx = ln |x ^4 + x ^3 | + 1 /x + C

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The function under integral is `( 4 x^2 + 2 x - 1 ) / ( x^3 + x^2 ) .` The degree of the denominator (3) is greater than the degree of the numerator (2), so there is no need for polynomial division.

The denominator is equal to...

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The function under integral is `( 4 x^2 + 2 x - 1 ) / ( x^3 + x^2 ) .` The degree of the denominator (3) is greater than the degree of the numerator (2), so there is no need for polynomial division.

The denominator is equal to `x^2 ( x + 1 ),` so the partial fraction decomposition has the form

`( 4 x^2 + 2 x - 1 ) / ( x^3 + x^2 ) = A / x^2 + B / x + C / ( x + 1 ) .`

To find the coefficients A, B, and C, multiply both sides by the denominator:

`4 x^2 + 2 x - 1 = A ( x + 1 ) + B x ( x + 1 ) + C x^2 .`

Then group like terms:

`4 = B + C, 2 = A + B , -1 = A .`

This way `A = -1 ,` `B = 2 - A = 3 ,` `C = 4 - B = 1 .`

Now we can easily integrate,

`int ( 4 x^2 + 2 x - 1 ) / ( x^3 + x^2 ) dx = int ( - 1 / x^2 + 3 / x + 1 / ( x + 1 ) ) dx =`

`= 1 / x + 3 ln | x | + ln | x + 1 | + C = 1 / x + ln | x^3 ( x + 1 ) | + C =`

` = 1 / x + ln | x^4 + x^3 | + C ,`

which is what we want.

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