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Lesson Worksheet: Integration by Partial Fractions with Quadratic Factors Mathematics

In this worksheet, we will practice using partial fractions to evaluate integrals of rational functions with irreducible quadratic factors.

Q1:

Use partial fractions to evaluate 𝑥𝑥(𝑥+1)d.

  • Alnln|𝑥|12𝑥+1+1(2𝑥+2)+𝐾
  • Blnln|𝑥|𝑥+1+12(𝑥+1)+𝐾
  • Clnln|𝑥|12𝑥+1+1(𝑥+1)+𝐾
  • Dlnln|𝑥|12𝑥+1+1(𝑥+1)+𝐾
  • Elnln|𝑥|12𝑥+11(2𝑥+2)+𝐾

Q2:

Use partial fractions to evaluate 2𝑠+2(𝑠+1)(𝑠1)𝑠d.

  • Atan𝑠(𝑠1)𝑠+1+𝐾
  • Btan𝑠+(𝑠1)(𝑠1)+𝐾
  • Ctan𝑠(𝑠1)(𝑠1)+𝐾
  • Dtan𝑠+(𝑠1)+(𝑠1)+𝐾
  • Etan𝑠+(𝑠+1)+(𝑠+1)+𝐾

Q3:

Use partial fractions to evaluate 8𝑥+8𝑥+2(4𝑥+1)𝑥d.

  • Atan2𝑥14𝑥+1+𝐾
  • Btan2𝑥+14𝑥+1+𝐾
  • Ctan𝑥14𝑥+1+𝐾
  • Dtan2𝑥24𝑥+1+𝐾
  • Etan2𝑥14𝑥+1+𝐾

Q4:

Use partial fractions to evaluate 𝑠+81𝑠(𝑠+9)𝑠d.

  • Aln|𝑠|+18(𝑠+9)+𝐾
  • Bln|𝑠|+9(𝑠+9)+𝐾
  • Cln|𝑠|18(𝑠+9)+𝐾
  • Dln|𝑠|+9(𝑠+9)+𝐾
  • Eln|𝑠|9(𝑠+9)+𝐾

Q5:

Use partial fractions to evaluate 𝜃4𝜃+2𝜃3𝜃+1(𝜃+1)𝜃d.

  • Atan𝜃+1𝜃+114(𝜃+1)+𝐾
  • Btan𝜃+2𝜃+114(𝜃+1)+𝐾
  • Ctan𝜃+2𝜃+114(𝜃+1)+𝐾
  • Dtan𝜃+2𝜃+1+14(𝜃+1)+𝐾
  • Etan𝜃+2𝜃+11(𝜃+1)+𝐾

Q6:

Use partial fractions to evaluate (𝑥+1)(3𝑥)+9𝑥+𝑥(9𝑥+1)(𝑥+1)𝑥tand.

  • Atanln(3𝑥)3+|𝑥+1|+1𝑥+1+𝐾
  • B(3𝑥)6|𝑥+1|1𝑥+1+𝐾tanln
  • Ctanln(3𝑥)6+|𝑥+1|+1𝑥+1+𝐾
  • D(3𝑥)6+|𝑥+1|+1𝑥+1+𝐾tanln
  • Etanln(3𝑥)6|𝑥+1|+1𝑥+1+𝐾

Q7:

Use partial fractions to evaluate 2𝑥+4(𝑥+1)(𝑥1)𝑥d.

  • Alntanln𝑥+1+𝑥2|𝑥1|+1𝑥1+𝐾
  • Blntanln𝑥+1+𝑥+2|𝑥1|1𝑥1+𝐾
  • Clntanln𝑥+1+𝑥2|𝑥1|1𝑥1+𝐾
  • Dlntanln𝑥+1+𝑥2|𝑥1|13(𝑥1)+𝐾
  • E12𝑥+1+𝑥2|𝑥1|1𝑥1+𝐾lntanln

Q8:

Use partial fractions to evaluate 𝑥𝑥+2𝑥1𝑥d.

  • A23|𝑥|+13𝑥+𝑥+1+32𝑥+13+𝐾lnlntan
  • B23|𝑥1|+13𝑥+𝑥+132𝑥+13+𝐾lnlntan
  • C23|𝑥1|+16𝑥+𝑥+1+32𝑥+13+𝐾lnlntan
  • D23|𝑥1|+16𝑥+𝑥+132𝑥+13+𝐾lnlntan
  • E23|𝑥1|+16𝑥+𝑥+132𝑥+13+𝐾lnlntan

Q9:

Use partial fractions to evaluate 𝑦+2𝑦+1(𝑦+1)𝑦d.

  • A2𝑦2𝑦+1+𝐾tan
  • Btan𝑦+1𝑦+1+𝐾
  • Ctan𝑦1𝑦+1+𝐾
  • Dtan𝑦1𝑦+1+𝐾
  • Etan𝑦1𝑦+1+𝐾

Q10:

Use partial fractions to evaluate 1𝑥+𝑥𝑥d.

  • Alnlnln|𝑥|13|𝑥+1|13||𝑥𝑥+1||+𝐾
  • Blnlnln|𝑥|16|𝑥+1|16||𝑥𝑥+1||+𝐾
  • Clnlnln|𝑥|+13|𝑥+1|+13||𝑥𝑥+1||+𝐾
  • Dlnlnln|𝑥|+23|𝑥+1|+23||𝑥𝑥+1||+𝐾
  • Elnlnln|𝑥|23|𝑥+1|23||𝑥𝑥+1||+𝐾

This lesson includes 2 additional questions for subscribers.

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