How do you integrate #int arctanx/((x^2)(x-1))dx# using partial fractions?

Answer 1
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Answer 2

To integrate ( \int \frac{\arctan(x)}{x^2(x-1)} , dx ) using partial fractions, first, express the integrand as a sum of partial fractions.

  1. Decompose the fraction ( \frac{\arctan(x)}{x^2(x-1)} ) into partial fractions.
  2. Find the constants by equating coefficients.
  3. Integrate each partial fraction separately.

The steps are as follows:

  1. Express ( \frac{\arctan(x)}{x^2(x-1)} ) as:

[ \frac{\arctan(x)}{x^2(x-1)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x-1} ]

  1. Multiply both sides by the denominator ( x^2(x-1) ) to clear the fractions:

[ \arctan(x) = A(x-1) + B(x)(x-1) + C(x^2) ]

  1. Plug in convenient values for ( x ) to find the constants ( A ), ( B ), and ( C ). A common approach is to choose values that make one or more terms disappear or simplify the equation.

  2. Once you have found the values of ( A ), ( B ), and ( C ), integrate each term separately.

  3. Finally, combine the integrals of the partial fractions to obtain the result.

Following these steps, you can integrate ( \frac{\arctan(x)}{x^2(x-1)} ) using partial fractions.

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Answer from HIX Tutor

When evaluating a one-sided 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 one-sided 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 one-sided 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 one-sided 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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