How do you integrate #int x/(x6) dx# using partial fractions?
Don't need to use partial fractions.
Step 2: Integrate
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To integrate the function ( \frac{x}{x6} ) using partial fractions, follow these steps:

Write the fraction as a sum of simpler fractions using partial fraction decomposition: ( \frac{x}{x6} = \frac{A}{x6} + \frac{B}{x} )

Multiply both sides by the common denominator ( (x6) \times x ) to clear the fractions: ( x = A \times x + B \times (x6) )

Substitute values for ( x ) that make the denominators zero to solve for ( A ) and ( B ).
 Substitute ( x = 0 ): ( 0 = 0 \times A + B \times (0  6) = 6B )
 Substitute ( x = 6 ): ( 6 = A \times 6 + 0 = 6A )

Solve the system of equations: ( A = 1 ) and ( B = 1 )

Rewrite the original integral with the partial fractions: ( \int \frac{x}{x6} dx = \int \frac{1}{x6} dx  \int \frac{1}{x} dx )

Integrate each term separately: ( \int \frac{1}{x6} dx = \lnx6 ) and ( \int \frac{1}{x} dx = \lnx )

Combine the results: ( \int \frac{x}{x6} dx = \lnx6  \lnx + C ) where ( C ) is the constant of integration.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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