Worksheet: Integration by Trigonometric Substitutions

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 ο„Έ(1βˆ’π‘₯)π‘₯π‘₯d.

  • A13π‘₯βˆ’15π‘₯+C
  • B15ο€βˆš1βˆ’π‘₯π‘₯+C
  • Cβˆ’15ο€βˆš1βˆ’π‘₯π‘₯+C
  • D13π‘₯+15π‘₯+C
  • Eβˆ’13π‘₯βˆ’15π‘₯+C

Q2:

Use a trigonometric substitution to evaluate ο„Έπ‘₯π‘₯(π‘₯βˆ’1)d, where π‘₯>1.

  • Aβˆ’14ο€Ώπ‘₯√π‘₯βˆ’1+C
  • B14ο€Ώπ‘₯√π‘₯βˆ’1+C
  • C13ο€Ώπ‘₯√π‘₯βˆ’1+C
  • Dο€Ώπ‘₯√π‘₯βˆ’1+C
  • Eβˆ’13ο€Ώπ‘₯√π‘₯βˆ’1+C

Q3:

Use a trigonometric substitution to evaluate ο„Έβˆšπ‘¦βˆ’49π‘¦π‘¦οŠ¨d, where 𝑦>7.

  • Aβˆ’7𝑦+C
  • B7ο€»βˆšπ‘¦βˆ’49βˆ’ο€»π‘¦7+secC
  • Cο€»βˆšπ‘¦βˆ’49βˆ’7𝑦7+secC
  • D7𝑦+C
  • EsecCοŠ±οŠ§οŠ¨π‘¦7βˆ’βˆšπ‘¦βˆ’49+

Q4:

Use a trigonometric substitution to evaluate ο„Έ2π‘₯√1βˆ’4π‘₯√d.

  • A√22
  • Bπœ‹4
  • Cπœ‹2
  • D√24
  • Eπœ‹8

Q5:

Use a trigonometric substitution to evaluate ο„Έβˆš1βˆ’(π‘₯)π‘₯π‘₯π‘₯lnlnd.

  • A√1βˆ’(π‘₯)βˆ’|||π‘₯1+√1βˆ’(π‘₯)|||+lnlnlnlnC
  • B√1βˆ’(π‘₯)+|||π‘₯1+√1βˆ’π‘₯|||+lnlnlnlnC
  • C√1βˆ’(π‘₯)+|||π‘₯1+√1βˆ’(π‘₯)|||+lnlnlnlnC
  • D√1+(π‘₯)+|||π‘₯1+√1βˆ’(π‘₯)|||+lnlnlnlnC
  • E√1βˆ’π‘₯+|||π‘₯1+√1βˆ’(π‘₯)|||+lnlnlnlnC

Q6:

Use a trigonometric substitution to evaluate ο„Έπ‘₯(π‘₯βˆ’1)d, where π‘₯>1.

  • Aβˆ’π‘₯√π‘₯βˆ’1+C
  • Bπ‘₯√π‘₯βˆ’1+C
  • Cβˆ’1√π‘₯βˆ’1+C
  • Dπ‘₯√π‘₯βˆ’1+C
  • Eβˆ’π‘₯√π‘₯βˆ’1+C

Q7:

Use a trigonometric substitution to evaluate ο„Έπ‘₯π‘₯√1+π‘₯dοŠͺ.

  • A12ο€»π‘₯βˆ’βˆš1+π‘₯+lnCοŠͺ
  • B12ο€»π‘₯+√1+π‘₯+lnCοŠͺ
  • C2ο€»π‘₯+√1+π‘₯+lnCοŠͺ
  • D2ο€»π‘₯+√1+π‘₯+lnCοŠͺ
  • E12ο€»π‘₯+√1+π‘₯+lnCοŠͺ

Q8:

Use a trigonometric substitution to evaluate ο„Έπ‘₯√9+π‘₯d.

  • AlnC|||9+π‘₯3+π‘₯3|||+
  • BlnC||||√9+π‘₯9||||+
  • ClnC||||√9+π‘₯9+π‘₯9||||+
  • DlnC||||√9+π‘₯3+π‘₯3||||+
  • ElnC||||√9+π‘₯3||||+

Q9:

Use a trigonometric substitution to evaluate ο„Έπ‘₯4+π‘₯d.

  • A12
  • B2
  • Cπœ‹4
  • Dπœ‹8
  • Eπœ‹4

Q10:

Use trigonometric substitution to evaluate ο„Έπ‘₯√1βˆ’π‘₯d.

  • AsinCπ‘₯+
  • BcosCπ‘₯+
  • CcosCπ‘₯+
  • DtanCπ‘₯+
  • EsinCπ‘₯+

Q11:

Use a trigonometric substitution to evaluate 𝑦1+𝑦d.

  • A12(𝑦)βˆ’π‘¦2+2𝑦+tanC
  • BsecC(𝑦)+
  • CtanC(𝑦)+
  • DsecC(2𝑦)+
  • E12(𝑦)+𝑦2+2𝑦+tanC

Q12:

Use a trigonometric substitution to evaluate ο„Έπ‘₯√4π‘₯βˆ’49d, where π‘₯>72.

  • A12||||√4π‘₯βˆ’497||||+lnC
  • B12||||√4π‘₯βˆ’497+2π‘₯7||||+lnC
  • ClnC||||√4π‘₯βˆ’497+4π‘₯7||||+
  • DlnC||||√4π‘₯βˆ’497+2π‘₯7||||+
  • E12||||√4π‘₯βˆ’497+4π‘₯7||||+lnC

Q13:

Use an appropriate substitution and then a trigonometric one to evaluate ο„Έο„ž4βˆ’π‘₯π‘₯π‘₯d.

  • A4ο€Ώβˆšπ‘₯2+√π‘₯√4βˆ’π‘₯+sinC
  • B4ο€Ώβˆšπ‘₯2+√4βˆ’π‘₯+sinC
  • ClnC√4βˆ’π‘₯2+
  • DlnC√π‘₯2+
  • E4ο€Ώβˆšπ‘₯2+√π‘₯√4βˆ’π‘₯+sinC

Q14:

Use a trigonometric substitution to evaluate ο„Έπ‘₯√9βˆ’π‘₯d.

  • Aπœ‹4
  • B12
  • C32
  • Dπœ‹3
  • Eπœ‹6

Q15:

Use an appropriate substitution followed by a trigonometric one to evaluate 𝑒𝑑(𝑒+1)lnlnd.

  • A75
  • B25
  • Cβˆ’15
  • D15
  • E712

Q16:

Use a trigonometric substitution to evaluate ο„Έβˆšπ‘¦βˆ’25π‘¦π‘¦οŠ¨οŠ©d, where 𝑦>5.

  • A110𝑦5+βˆšπ‘¦βˆ’252𝑦+secC
  • B110𝑦5ο‡βˆ’βˆšπ‘¦βˆ’25𝑦+secC
  • C52𝑦5ο‡βˆ’25βˆšπ‘¦βˆ’252𝑦+secC
  • D110𝑦5ο‡βˆ’βˆšπ‘¦βˆ’252𝑦+secC
  • EsecCοŠ±οŠ§οŠ¨οŠ¨ο€»π‘¦5ο‡βˆ’5βˆšπ‘¦βˆ’25𝑦+

Q17:

Use a trigonometric substitution to evaluate 𝑣𝑣(1βˆ’π‘£)d.

  • Aο€Ώπ‘£βˆš1βˆ’π‘£ο‹+C
  • B12𝑣+C
  • Cβˆ’12𝑣+C
  • D13ο€Ώπ‘£βˆš1βˆ’π‘£ο‹+C
  • Eβˆ’13ο€Ώπ‘£βˆš1βˆ’π‘£ο‹+C

Q18:

Use a trigonometric substitution to evaluate ο„Έπ‘₯π‘₯√π‘₯βˆ’1d, where π‘₯>1.

  • AlnC||√π‘₯βˆ’1||βˆ’π‘₯βˆ’12+
  • B√π‘₯βˆ’1π‘₯+C
  • Cβˆ’π‘₯√π‘₯βˆ’1+C
  • D1π‘₯+C
  • Eβˆ’βˆšπ‘₯βˆ’1π‘₯+C

Q19:

Use a trigonometric substitution to evaluate ο„Έ4π‘₯π‘₯(1βˆ’π‘₯)√d.

  • A4√3+4πœ‹3
  • B4√3βˆ’4πœ‹3
  • C4√33+2πœ‹3
  • D2√3βˆ’2πœ‹3
  • E4√3βˆ’2πœ‹3

Q20:

Use a trigonometric substitution to evaluate ο„Έπ‘¦π‘¦βˆšπ‘¦βˆ’1d.

  • AlnC𝑦+1βˆšπ‘¦βˆ’1+1𝑦+
  • BtanC|𝑦|+
  • ClnC𝑦+1βˆšπ‘¦βˆ’1βˆ’1𝑦+
  • DsecC|𝑦|+
  • EsecC|𝑦|+

Q21:

Use a trigonometric substitution to evaluate ο„Έ3π‘₯√1+9π‘₯d.

  • AlnC||1+9π‘₯+3π‘₯||+
  • BlnC||√1+9π‘₯+3π‘₯||+
  • ClnC||||√9+π‘₯9||||+
  • DlnC||√1+9π‘₯||+
  • ElnC||||√9+π‘₯3+π‘₯3||||+

Q22:

Use a trigonometric substitution to evaluate ο„Έπ‘₯√π‘₯+4π‘₯d.

  • Aο€»βˆšπ‘₯+43βˆ’8√π‘₯+4+C
  • Bο€»βˆšπ‘₯+43βˆ’4√π‘₯+4+C
  • C√π‘₯+42+8√π‘₯+4+C
  • Dο€»βˆšπ‘₯+43+4√π‘₯+4+C
  • Eο€»βˆšπ‘₯+42+4√π‘₯+4+C

Q23:

Use a trigonometric substitution to evaluate ο„Έπ‘₯(4βˆ’π‘₯)d.

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

Q24:

Use a trigonometric substitution to evaluate ο„Έπ‘₯4+π‘₯π‘₯d.

  • Aπ‘₯βˆ’2ο€»π‘₯2+tanC
  • Bπ‘₯+2ο€»π‘₯2+tanC
  • C12π‘₯2βˆ’π‘₯π‘₯+4+tanC
  • D12π‘₯2+π‘₯π‘₯+4+tanC
  • Eπ‘₯βˆ’ο€»π‘₯2+tanC

Q25:

Use an appropriate substitution followed by a trigonometric one to evaluate ο„Έ2π‘‘βˆšπ‘‘+4π‘‘βˆšπ‘‘οŽ οŽ£οŽ οŽ οŽ‘d.

  • A23
  • Bπœ‹6
  • C13
  • Dπœ‹12
  • Eπœ‹2

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