How do you integrate #int arctanx/((x^2)(x-1))dx# using partial fractions?
You don't, sorry !
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To integrate ( \int \frac{\arctan(x)}{x^2(x-1)} , dx ) using partial fractions, first, express the integrand as a sum of partial fractions.
- Decompose the fraction ( \frac{\arctan(x)}{x^2(x-1)} ) into partial fractions.
- Find the constants by equating coefficients.
- Integrate each partial fraction separately.
The steps are as follows:
- 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} ]
- 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) ]
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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.
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Once you have found the values of ( A ), ( B ), and ( C ), integrate each term separately.
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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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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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