Integrating with Partial Fractions (Edexcel A-Level Further Mathematics): Quizzes

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Integrating with Partial Fractions
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What is the partial fraction form of P(x)(x−a)(x−b)\frac{P(x)}{(x-a)(x-b)} with distinct linear factors?

Ax−a+Bx−b\frac{A}{x-a} + \frac{B}{x-b}

What is the partial fraction form of P(x)(x−a)2\frac{P(x)}{(x-a)^2} with repeated linear factors?

Ax−a+B(x−a)2\frac{A}{x-a} + \frac{B}{(x-a)^2}

What is the partial fraction form for an irreducible quadratic factor ax2+bx+cax^2+bx+c?

Ax+Bax2+bx+c\frac{Ax+B}{ax^2+bx+c}

What is the first step when integrating using partial fractions?

Decompose the fraction

How are constants determined in partial fraction decomposition?

Equate numerators or substitute values

What is ∫1xdx\int \frac{1}{x} dx equal to?

ln⁡∣x∣+C\ln|x| + C

What is ∫xx2+a2dx\int \frac{x}{x^2+a^2} dx equal to?

12ln⁡(x2+a2)+C\frac{1}{2} \ln(x^2+a^2) + C

What must be ensured before setting up partial fractions?

Denominator is properly factored

What must always be included in indefinite integrals?

Constant of integration +C+C

What is ∫1x2+a2dx\int \frac{1}{x^2+a^2} dx equal to?

1aarctan⁡(xa)+C\frac{1}{a} \arctan(\frac{x}{a}) + C

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