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Notely Maths 12
Class 12 Maths
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05_Ex 3.5

Exercise Questions

QuestionsQuestion Links
Q1. ∫3x+1x2−x−6dx\int \frac{3 x+1}{x^{2}-x-6} d x3.5 Q-1
Q2. ∫5x+8(x+3)(2x−1)dx\int \frac{5 x+8}{(x+3)(2 x-1)} d x3.5 Q-2
Q3. ∫x2+3x−34x2+2x−15dx\int \frac{x^{2}+3 x-34}{x^{2}+2 x-15} d x3.5 Q-3
Q4. ∫(a−b)x(x−a)(x−b)dxdx,(a>b)\int \frac{(a-b) x}{(x-a)(x-b)} d x \quad d x,(a>b)3.5 Q-4
Q5. ∫3−x1−x−6x2dx\int \frac{3-\mathrm{x}}{1-\mathrm{x}-6 \mathrm{x}^{2}} \mathrm{dx}3.5 Q-5
Q6. ∫2xx2−a2dx\int \frac{2 x}{x^{2}-a^{2}} d x3.5 Q-6
Q7. ∫16x2+5x−4dx\int \frac{1}{6 x^{2}+5 x-4} d x3.5 Q-7
Q8. ∫2x3−3x2−x−72x2−3x−2dx\int \frac{2 x^{3}-3 x^{2}-x-7}{2 x^{2}-3 x-2} d x3.5 Q-8
Q9. ∫3x2−12x+11(x−1)(x−2)(x−3)dx\int \frac{3 x^{2}-12 x+11}{(x-1)(x-2)(x-3)} d x3.5 Q-9
Q10. ∫2x−1x(x−1)(x−3)dx\int \frac{2 x-1}{x(x-1)(x-3)} d x3.5 Q-10
Q11. ∫5x2+9x+6(x2−1)(2x+3)dx\int \frac{5 x^{2}+9 x+6}{\left(x^{2}-1\right)(2 x+3)} d x3.5 Q-11
Q12. ∫4+7x(1+x)2(2+3x)dx\int \frac{4+7 x}{(1+\mathrm{x})^{2}(2+3 \mathrm{x})} \mathrm{dx}3.5 Q-12
Q13. ∫2x2(x−1)2(x+1)dx\int \frac{2 x^{2}}{(\mathrm{x}-1)^{2}(\mathrm{x}+1)} \mathrm{dx}3.5 Q-13
Q14. ∫1(x+1)2(x−1)dx\int \frac{1}{(x+1)^{2}(x-1)} d x3.5 Q-14
Q15. ∫x+4x3−3x2+4dx\int \frac{\mathrm{x}+4}{\mathrm{x}^{3}-3 \mathrm{x}^{2}+4} \mathrm{dx}3.5 Q-15
Q16. ∫x2−6x2+25(x+1)2(x−2)2dx\int \frac{x^{2}-6 x^{2}+25}{(x+1)^{2}(x-2)^{2}} d x3.5 Q-16
Q17. ∫x3+22x2+14x−17(x−3)(x+2)3dx\int \frac{x^{3}+22 x^{2}+14 x-17}{(x-3)(x+2)^{3}} d x3.5 Q-17
Q18. ∫x−2(x+1)(x2+1)dx\int \frac{x-2}{(x+1)\left(x^{2}+1\right)} d x3.5 Q-18
Q19. ∫x(x−1)(x2+1)dx\int \frac{x}{(x-1)\left(x^{2}+1\right)} d x3.5 Q-19
Q20. ∫9x−7(x+3)(x2+1)dx\int \frac{9 x-7}{(x+3)\left(x^{2}+1\right)} d x3.5 Q-20
Q21. ∫1+4x(x−3)(x2+4)dx\int \frac{1+4 x}{(x-3)\left(x^{2}+4\right)} d x3.5 Q-21
Q22. ∫12x3+8dx\int \frac{12}{\mathrm{x}^{3}+8} \mathrm{dx}3.5 Q-22
Q23. ∫9x+6(x2−8)dx\int \frac{9 x+6}{\left(x^{2}-8\right)} d x3.5 Q-23
Q24. ∫2x2+5x+3(x−1)2(x2+4)dx\int \frac{2 x^{2}+5 x+3}{(x-1)^{2}\left(x^{2}+4\right)} d x3.5 Q-24
Q25. ∫2x2−x−7(x+2)2(x2+x+1)dx\int \frac{2 x^{2}-\mathrm{x}-7}{(\mathrm{x}+2)^{2}\left(\mathrm{x}^{2}+\mathrm{x}+1\right)} \mathrm{dx}3.5 Q-25
Q26. ∫3x+1(4x2+1)(x2−x+1)dx\int \frac{3 x+1}{\left(4 x^{2}+1\right)\left(x^{2}-x+1\right)} d x3.5 Q-26
Q27. ∫4x+1(x2+4)(x2+4x+5)dx\int \frac{4 \mathrm{x}+1}{\left(\mathrm{x}^{2}+4\right)\left(\mathrm{x}^{2}+4 \mathrm{x}+5\right)} \mathrm{dx}3.5 Q-27
Q28. ∫6a2(x2+a2)(x2+4a2)dx\int \frac{6 a^{2}}{\left(x^{2}+a^{2}\right)\left(x^{2}+4 a^{2}\right)} d x3.5 Q-28
Q29. ∫2x2−2(x4+x2+1)dx\int \frac{2 \mathrm{x}^{2}-2}{\left(\mathrm{x}^{4}+\mathrm{x}^{2}+1\right)} d x3.5 Q-29
Q30. ∫3x−8(x2+x+2)(x2−x+2)dx\int \frac{3 \mathrm{x}-8}{\left(\mathrm{x}^{2}+\mathrm{x}+2\right)\left(x^{2}-x+2\right)} d x3.5 Q-30
Q31. ∫3x3+4x2+9x+5(x2+x+1)(x2+2x+3)dx\int \frac{3 x^{3}+4 x^{2}+9 x+5}{\left(\mathrm{x}^{2}+\mathrm{x}+1\right)\left(x^{2}+2 x+3\right)} d x3.5 Q-31

Overview

This exercise focuses on integration using partial fractions, which is a technique used to break down rational functions into simpler fractions that can be integrated more easily. Partial fractions help solve integrals of rational functions, where the numerator and denominator are polynomials.

Key Concepts

  • Partial Fractions: A method of decomposing a rational function into a sum of simpler fractions.
  • Rational Function: A function that is the ratio of two polynomials.
  • Linear Factors: When the denominator factors into linear terms, partial fractions can be used to express the integrand in a simpler form.

Important Formulas

  • General Formula for Partial Fractions:
    Ax+b+Bx−c\frac{A}{x + b} + \frac{B}{x - c}
    This is used when the denominator factors into linear terms.

  • Integral of 1x\frac{1}{x}:
    ∫1xdx=ln⁡∣x∣+C\int \frac{1}{x} dx = \ln |x| + C

More details are discussed in the Questions

Tips and Tricks

  • Factor First: Always factor the denominator fully before applying partial fractions.
  • Substitute Values for Coefficients: When solving for the constants in partial fractions, substitute convenient values of xx (like x=−2x = -2 or x=0x = 0) to quickly solve for the unknowns.
  • Simplify Step-by-Step: Work through each term in the partial fraction decomposition methodically to avoid errors.

Summary

In this exercise, we learned how to integrate rational functions by decomposing them into partial fractions. This method simplifies complex rational expressions into simpler ones that are easier to integrate. The integral of each simpler fraction is then solved using basic integration formulas, such as those for ln⁡∣x∣\ln |x|.

Reference

By Sir Shahzad Sair:

By Great Science Academy: