Integration by Partial Fraction

Practice Questions

1.\(\int_{}^{}{\frac{7x}{\left( x+2 \right)\left( 2x-3 \right)}}\; dx\)

Ans : \(2\log\left( x\; +\; 2 \right)\; +\; \frac{3}{2}\log\left( 2x\; -\; 3 \right)\; +\; c\)

2. \(\int_{}^{}{\frac{5x\; -\; 11}{2x^{2}\; +\; x\; -{\; 6}}}\; dx\)

Ans : \(-\frac{1}{2}\; \log \; \left( 2x\; -\; 3 \right)\; +\; 3\; \log \; \left( x\; +\; 2 \right)\; +\; c\)

3. \(\int_{}^{}{\frac{3x}{\left( x\; -\; 2 \right)\left( x\; +\; 1 \right)}\;dx}\)

Ans : \(2\log\left( x\; -\; 2 \right)\; +\; \log\left( x\; +\; 1 \right)\; +\; c\)

4. \(\int_{}^{}{\frac{1}{\left( x\; +\; 1 \right)\left( 2x\; +\; 1 \right)}\; dx}\)

Ans : \(\log\left( 2x\; +\; 1 \right)\; -\; \log \left( x\; +\; 1 \right)\; +\; c\)

5. \(\int_{}^{}{\frac{x}{\left( x\; -\; 1 \right)\left( 2x\; +\; 1 \right)}}\;dx\)

Ans : \(\frac{1}{3}\left\{ \log\left( x\; -\; 1 \right)\; +\; \frac{1}{2}\log\left( 2x\; -\; 1 \right) \right\}\; +\; c\; \)

6.\(\int_{}^{}{\frac{1}{x^{2}\; -\; 3x\; +\; 2}}\;dx\)

Ans : \(\log \left( x\; -\; 2 \right)\; -\; \log \left( x\; -\; 1 \right)\; +\; c\)

7. \(\int_{}^{}{\frac{x\; +\; 2}{\left( x\; -\; 2 \right)\left( 2x\; +\; 3 \right)}}\;dx\)

Ans : \(\frac{4}{7}\; \log\left( x\; -\; 2 \right)\; -\; \frac{1}{14}\; \log\left( 2x\; +\; 3 \right)\; +\; c\)

8. \(\int_{}^{}{\frac{3x\; -\; 20}{x^{2}\; +\; 3x\; -\; 10}}\;dx\)

Ans : \(5\; \log\left( x\; +\; 5 \right)\; -\; 2\log \left( x\; -\; 2 \right)\; +\; c\)

9. \(\int_{}^{}{\frac{23x\; -\; 11x^{2}\; }{\left( 2x\; -\; 1 \right)\left( 9\; -\; x^{2} \right)}}\;dx\)

Ans : \(\frac{1}{2}\log \left( 2x\; -\; 1 \right)\; +\; 4\log\left( 3\; +\; x \right)\; +\; \log\left( 3\; -\; x \right)\; +\; c\)

10. \(\int_{}^{}{\frac{3x^{2}\; +\; x\; -\; 2}{\left( x\; -\; 2 \right)^{2}\; \left( 1\; -\; 2x \right)}\; \;dx}\)

Ans : \(-\frac{5}{3}\log \left( x\; -\; 2 \right)\; +\; \frac{4}{\left( x\; -\; 2 \right)}\; +\; \frac{1}{6}\log \left( 1\; -\; 2x \right)\; +\; c\)

11. \(\int_{}^{}{\frac{x}{\left( x\; -\; 1 \right)^{2}\; \left( x\; +\; 2 \right)} \;dx}\)

Ans : \(\frac{2}{9}\; \log \left( x\; -\; 1 \right)\; -\; \frac{2}{9}\; \log \left( x\; +\; 2 \right)\; -\; \frac{1}{3\left( x\; -\; 1 \right)}\; +\; c\; \)

12. \(\int_{}^{}{\frac{x^{2}}{\left( x\; +\; 1 \right)\; \left( x\; +\; 2 \right)\; \left( x\; +\; 3 \right)}\; \;dx}\)

Ans : \(\frac{1}{2}\; \log\left( x\; +\; 1 \right)\; -\; 4\; \log \left( x\; +\; 2 \right)\; +\; \frac{9}{2}\; \log \left( x\; +\; 3 \right)\; +\; c\)

13. \(\int_{}^{}{\frac{\left( x\; -\; 1 \right)}{\left( x\; +\; 1 \right)\left( x^{2}\; +\; 1 \right)}}\;dx\)

Ans : \(-\log\left( x\; +\; 1 \right)\; +\; \frac{1}{2}\; \log \left( x^{2}\; +\; 1 \right)\; +\; c\)

14. \(\int_{}^{}{\frac{x^{3}\; -\; 5x}{\left( x^{2}\; -\; 9 \right)\; \left( x^{2}\; +\; 1 \right)}\; \;dx}\)

Ans. \(\frac{1}{5}\log\left( x^{2}\; -\; 9 \right)\; +\; \frac{3}{10}\log \left( x^{2}\; +\; 1 \right)\; +\; c\)

15. \(\int_{}^{}{\frac{2x}{\left( x^{2}\; +\; 1 \right)\left( x^{2}\; +\; 3 \right)}}\; dx\)

Ans : \(\frac{1}{2}\log\left( x^{2}\; +\; 1 \right)\; -\; \frac{1}{2}\log\left( x^{2}\; +\; 3 \right)\; +\; c\)

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