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Partial fraction decomposition practice problems

Split a rational function into simpler fractions, and integrate it term by term. 10 problems with worked solutions; in 10 of them every equation is proved by a computer algebra system.

Find the partial fraction decomposition of \( \displaystyle \frac{x^{2} - 10 x + 8}{\left(x + 2\right) \left(x^{2} + 4\right)} \).

Problem 4.390medium✓ Every equation proved

Find the partial fraction decomposition of \( \displaystyle \frac{6 x - 4}{\left(x - 3\right) \left(x + 4\right)} \).

Problem 4.393medium✓ Nihil obstat

Find the partial fraction decomposition of \( \displaystyle \frac{5 x - 4}{x^{3} + x^{2} + 2 x + 2} \).

Problem 4.395medium✓ Nihil obstat

Find the partial fraction decomposition of \( \displaystyle \frac{- 7 x - 7}{x^{2} - x - 12} \).

Problem 4.396medium✓ Nihil obstat

Find the partial fraction decomposition of \( \displaystyle \frac{- 9 x^{2} + 4 x - 12}{x^{3} - 2 x^{2} + 4 x - 8} \).

Problem 4.398medium✓ Nihil obstat

Evaluate \( \displaystyle \int \frac{x - 4}{\left(x - 3\right) \left(x - 2\right)}\, dx \) using partial fractions.

Problem 4.391hard✓ Every equation proved

Evaluate \( \displaystyle \int \frac{- 9 x^{2} + 32 x - 20}{\left(x - 4\right) \left(x - 1\right)^{2}}\, dx \) using partial fractions.

Problem 4.392hard✓ Every equation proved

Evaluate \( \displaystyle \int \frac{- x^{2} - 10 x + 32}{x^{3} - 2 x^{2} - 4 x + 8}\, dx \) using partial fractions.

Problem 4.394hard✓ Every equation proved

Evaluate \( \displaystyle \int \frac{4 x^{2} - 12 x + 13}{\left(x - 3\right) \left(x^{2} + 4\right)}\, dx \) using partial fractions.

Problem 4.397hard✓ Nihil obstat

Evaluate \( \displaystyle \int \frac{x^{2} + 26 x - 80}{x^{3} - 3 x^{2} - 16 x + 48}\, dx \) using partial fractions.

Problem 4.399hard✓ Nihil obstat