Worksheet: Integration by Trigonometric Substitutions

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In this worksheet, we will practice using trigonometric substitutions to evaluate integrals containing radicals of the form √(x² ± a²).

Q1:

Use a trigonometric substitution to evaluate (1−𝑥)𝑥𝑥d.

  • A13𝑥−15𝑥+C
  • B15√1−𝑥𝑥+C
  • C−15√1−𝑥𝑥+C
  • D13𝑥+15𝑥+C
  • E−13𝑥−15𝑥+C

Q2:

Use a trigonometric substitution to evaluate 𝑥𝑥(𝑥−1)d, where 𝑥>1.

  • A−14𝑥√𝑥−1+C
  • B14𝑥√𝑥−1+C
  • C13𝑥√𝑥−1+C
  • D𝑥√𝑥−1+C
  • E−13𝑥√𝑥−1+C

Q3:

Use a trigonometric substitution to evaluate √𝑦−49𝑦𝑦d, where 𝑦>7.

  • A−7𝑦+C
  • B7√𝑦−49−𝑦7+secC
  • C√𝑦−49−7𝑦7+secC
  • D7𝑦+C
  • EsecC𝑦7−√𝑦−49+

Q4:

Use a trigonometric substitution to evaluate 2𝑥√1−4𝑥√d.

  • A√22
  • B𝜋4
  • C𝜋2
  • D√24
  • E𝜋8

Q5:

Use a trigonometric substitution to evaluate √1−(𝑥)𝑥𝑥𝑥lnlnd.

  • A√1−(𝑥)−|||𝑥1+√1−(𝑥)|||+lnlnlnlnC
  • B√1−(𝑥)+|||𝑥1+√1−𝑥|||+lnlnlnlnC
  • C√1−(𝑥)+|||𝑥1+√1−(𝑥)|||+lnlnlnlnC
  • D√1+(𝑥)+|||𝑥1+√1−(𝑥)|||+lnlnlnlnC
  • E√1−𝑥+|||𝑥1+√1−(𝑥)|||+lnlnlnlnC

Q6:

Use a trigonometric substitution to evaluate 𝑥(𝑥−1)d, where 𝑥>1.

  • A−𝑥√𝑥−1+C
  • B𝑥√𝑥−1+C
  • C−1√𝑥−1+C
  • D𝑥√𝑥−1+C
  • E−𝑥√𝑥−1+C

Q7:

Use a trigonometric substitution to evaluate 𝑥𝑥√1+𝑥d.

  • A12𝑥−√1+𝑥+lnC
  • B12𝑥+√1+𝑥+lnC
  • C2𝑥+√1+𝑥+lnC
  • D2𝑥+√1+𝑥+lnC
  • E12𝑥+√1+𝑥+lnC

Q8:

Use a trigonometric substitution to evaluate 𝑥√9+𝑥d.

  • AlnC|||9+𝑥3+𝑥3|||+
  • BlnC||||√9+𝑥9||||+
  • ClnC||||√9+𝑥9+𝑥9||||+
  • DlnC||||√9+𝑥3+𝑥3||||+
  • ElnC||||√9+𝑥3||||+

Q9:

Use a trigonometric substitution to evaluate 𝑥4+𝑥d.

  • A12
  • B2
  • C𝜋4
  • D𝜋8
  • E𝜋4

Q10:

Use trigonometric substitution to evaluate 𝑥√1−𝑥d.

  • AsinC𝑥+
  • BcosC𝑥+
  • CcosC𝑥+
  • DtanC𝑥+
  • EsinC𝑥+

Q11:

Use a trigonometric substitution to evaluate 𝑦1+𝑦d.

  • A12(𝑦)−𝑦2+2𝑦+tanC
  • BsecC(𝑦)+
  • CtanC(𝑦)+
  • DsecC(2𝑦)+
  • E12(𝑦)+𝑦2+2𝑦+tanC

Q12:

Use a trigonometric substitution to evaluate 𝑥√4𝑥−49d, where 𝑥>72.

  • A12||||√4𝑥−497||||+lnC
  • B12||||√4𝑥−497+2𝑥7||||+lnC
  • ClnC||||√4𝑥−497+4𝑥7||||+
  • DlnC||||√4𝑥−497+2𝑥7||||+
  • E12||||√4𝑥−497+4𝑥7||||+lnC

Q13:

Use an appropriate substitution and then a trigonometric one to evaluate 4−𝑥𝑥𝑥d.

  • A4√𝑥2+√𝑥√4−𝑥+sinC
  • B4√𝑥2+√4−𝑥+sinC
  • ClnC√4−𝑥2+
  • DlnC√𝑥2+
  • E4√𝑥2+√𝑥√4−𝑥+sinC

Q14:

Use a trigonometric substitution to evaluate 𝑥√9−𝑥d.

  • A𝜋4
  • B12
  • C32
  • D𝜋3
  • E𝜋6

Q15:

Use an appropriate substitution followed by a trigonometric one to evaluate 𝑒𝑡(𝑒+1)lnlnd.

  • A75
  • B25
  • C−15
  • D15
  • E712

Q16:

Use a trigonometric substitution to evaluate √𝑦−25𝑦𝑦d, where 𝑦>5.

  • A110𝑦5+√𝑦−252𝑦+secC
  • B110𝑦5−√𝑦−25𝑦+secC
  • C52𝑦5−25√𝑦−252𝑦+secC
  • D110𝑦5−√𝑦−252𝑦+secC
  • EsecC𝑦5−5√𝑦−25𝑦+

Q17:

Use a trigonometric substitution to evaluate 𝑣𝑣(1−𝑣)d.

  • A𝑣√1−𝑣+C
  • B12𝑣+C
  • C−12𝑣+C
  • D13𝑣√1−𝑣+C
  • E−13𝑣√1−𝑣+C

Q18:

Use a trigonometric substitution to evaluate 𝑥𝑥√𝑥−1d, where 𝑥>1.

  • AlnC||√𝑥−1||−𝑥−12+
  • B√𝑥−1𝑥+C
  • C−𝑥√𝑥−1+C
  • D1𝑥+C
  • E−√𝑥−1𝑥+C

Q19:

Use a trigonometric substitution to evaluate 4𝑥𝑥(1−𝑥)√d.

  • A4√3+4𝜋3
  • B4√3−4𝜋3
  • C4√33+2𝜋3
  • D2√3−2𝜋3
  • E4√3−2𝜋3

Q20:

Use a trigonometric substitution to evaluate 𝑦𝑦√𝑦−1d.

  • AlnC𝑦+1√𝑦−1+1𝑦+
  • BtanC|𝑦|+
  • ClnC𝑦+1√𝑦−1−1𝑦+
  • DsecC|𝑦|+
  • EsecC|𝑦|+

Q21:

Use a trigonometric substitution to evaluate 3𝑥√1+9𝑥d.

  • AlnC||1+9𝑥+3𝑥||+
  • BlnC||√1+9𝑥+3𝑥||+
  • ClnC||||√9+𝑥9||||+
  • DlnC||√1+9𝑥||+
  • ElnC||||√9+𝑥3+𝑥3||||+

Q22:

Use a trigonometric substitution to evaluate 𝑥√𝑥+4𝑥d.

  • A√𝑥+43−8√𝑥+4+C
  • B√𝑥+43−4√𝑥+4+C
  • C√𝑥+42+8√𝑥+4+C
  • D√𝑥+43+4√𝑥+4+C
  • E√𝑥+42+4√𝑥+4+C

Q23:

Use a trigonometric substitution to evaluate 𝑥(4−𝑥)d.

  • A2√3
  • B34√3
  • C12√3
  • D14√3
  • E1√3

Q24:

Use a trigonometric substitution to evaluate 𝑥4+𝑥𝑥d.

  • A𝑥−2𝑥2+tanC
  • B𝑥+2𝑥2+tanC
  • C12𝑥2−𝑥𝑥+4+tanC
  • D12𝑥2+𝑥𝑥+4+tanC
  • E𝑥−𝑥2+tanC

Q25:

Use an appropriate substitution followed by a trigonometric one to evaluate 2𝑡√𝑡+4𝑡√𝑡d.

  • A23
  • B𝜋6
  • C13
  • D𝜋12
  • E𝜋2

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