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Worksheet: Trigonometric Substitutions

In this worksheet, we will practice evaluating integrals using trigonometric substitutions.

Q1:

Use a trigonometric substitution to evaluate ο„Έ ( 1 βˆ’ π‘₯ ) π‘₯ π‘₯ 2 6 3 2 d .

  • A 1 3 π‘₯ βˆ’ 1 5 π‘₯ + 3 5 C
  • B 1 5  √ 1 βˆ’ π‘₯ π‘₯  + 2 5 C
  • C 1 3 π‘₯ + 1 5 π‘₯ + 3 5 C
  • D βˆ’ 1 5  √ 1 βˆ’ π‘₯ π‘₯  + 2 5 C
  • E βˆ’ 1 3 π‘₯ βˆ’ 1 5 π‘₯ + 3 5 C

Q2:

Use a trigonometric substitution to evaluate ο„Έ π‘₯ π‘₯ ( π‘₯ βˆ’ 1 ) 2 2 d 5 2 , where π‘₯ > 1 .

  • A ο€Ώ π‘₯ √ π‘₯ βˆ’ 1  + 2 3 C
  • B 1 3 ο€Ώ π‘₯ √ π‘₯ βˆ’ 1  + 2 3 C
  • C 1 4 ο€Ώ π‘₯ √ π‘₯ βˆ’ 1  + 2 2 C
  • D βˆ’ 1 3 ο€Ώ π‘₯ √ π‘₯ βˆ’ 1  + 2 3 C
  • E βˆ’ 1 4 ο€Ώ π‘₯ √ π‘₯ βˆ’ 1  + 2 2 C

Q3:

Use a trigonometric substitution to evaluate ο„Έ √ 𝑦 βˆ’ 4 9 𝑦 𝑦 2 d , where 𝑦 > 7 .

  • A 7 𝑦 + C
  • B s e c C βˆ’ 1 2 𝑦 7 βˆ’ √ 𝑦 βˆ’ 4 9 +
  • C βˆ’ 7 𝑦 + C
  • D ο€» √ 𝑦 βˆ’ 4 9 βˆ’ 7 ο€» 𝑦 7   + 2 βˆ’ 1 s e c C
  • E 7 ο€» √ 𝑦 βˆ’ 4 9 βˆ’ ο€» 𝑦 7   + 2 βˆ’ 1 s e c C

Q4:

Use a trigonometric substitution to evaluate ο„Έ 2 π‘₯ √ 1 βˆ’ 4 π‘₯ 1 2 √ 2 0 2 d .

  • A √ 2 4
  • B πœ‹ 2
  • C √ 2 2
  • D πœ‹ 4
  • E πœ‹ 8

Q5:

Use a trigonometric substitution to evaluate ο„Έ √ 1 βˆ’ ( π‘₯ ) π‘₯ π‘₯ π‘₯ l n l n d 2 .

  • A √ 1 βˆ’ π‘₯ + | | | π‘₯ 1 + √ 1 βˆ’ ( π‘₯ ) | | | + l n l n l n l n C 2
  • B √ 1 + ( π‘₯ ) + | | | π‘₯ 1 + √ 1 βˆ’ ( π‘₯ ) | | | + l n l n l n l n C 2 2
  • C √ 1 βˆ’ ( π‘₯ ) + | | | π‘₯ 1 + √ 1 βˆ’ π‘₯ | | | + l n l n l n l n C 2
  • D √ 1 βˆ’ ( π‘₯ ) + | | | π‘₯ 1 + √ 1 βˆ’ ( π‘₯ ) | | | + l n l n l n l n C 2 2
  • E √ 1 βˆ’ ( π‘₯ ) βˆ’ | | | π‘₯ 1 + √ 1 βˆ’ ( π‘₯ ) | | | + l n l n l n l n C 2 2

Q6:

Use a trigonometric substitution to evaluate ο„Έ π‘₯ ( π‘₯ βˆ’ 1 ) d 2 3 2 , where π‘₯ > 1 .

  • A βˆ’ π‘₯ √ π‘₯ βˆ’ 1 + 2 2 C
  • B π‘₯ √ π‘₯ βˆ’ 1 + 2 C
  • C π‘₯ √ π‘₯ βˆ’ 1 + 2 2 C
  • D βˆ’ π‘₯ √ π‘₯ βˆ’ 1 + 2 C
  • E βˆ’ 1 √ π‘₯ βˆ’ 1 + 2 C

Q7:

Use a trigonometric substitution to evaluate ο„Έ π‘₯ π‘₯ √ 1 + π‘₯ d 4 .

  • A 1 2 ο€» π‘₯ βˆ’ √ 1 + π‘₯  + l n C 2 4
  • B 1 2 ο€» π‘₯ + √ 1 + π‘₯  + l n C 4
  • C 2 ο€» π‘₯ + √ 1 + π‘₯  + l n C 4
  • D 1 2 ο€» π‘₯ + √ 1 + π‘₯  + l n C 2 4
  • E 2 ο€» π‘₯ + √ 1 + π‘₯  + l n C 2 4

Q8:

Use a trigonometric substitution to evaluate ο„Έ π‘₯ √ 9 + π‘₯ d 2 .

  • A l n C | | | | √ 9 + π‘₯ 9 + π‘₯ 9 | | | | + 2
  • B l n C | | | | √ 9 + π‘₯ 3 | | | | + 2
  • C l n C | | | | √ 9 + π‘₯ 9 | | | | + 2
  • D l n C | | | | √ 9 + π‘₯ 3 + π‘₯ 3 | | | | + 2
  • E l n C | | | 9 + π‘₯ 3 + π‘₯ 3 | | | + 2

Q9:

Use a trigonometric substitution to evaluate .

  • A2
  • B
  • C
  • D
  • E

Q10:

Use trigonometric substitution to evaluate ο„Έ π‘₯ √ 1 βˆ’ π‘₯ d 2 .

  • A t a n C βˆ’ 1 π‘₯ +
  • B c o s C βˆ’ 1 π‘₯ +
  • C s i n C π‘₯ + βˆ’ 1
  • D s i n C βˆ’ 1 π‘₯ +
  • E c o s C π‘₯ + βˆ’ 1

Q11:

Use a trigonometric substitution to evaluate ο„Έ 𝑦 1 + 𝑦 d 2 .

  • A 1 2 ( 𝑦 ) + 𝑦 2 + 2 𝑦 + t a n C βˆ’ 1 2
  • B s e c C ( 𝑦 ) +
  • C 1 2 ( 𝑦 ) βˆ’ 𝑦 2 + 2 𝑦 + t a n C βˆ’ 1 2
  • D t a n C βˆ’ 1 ( 𝑦 ) +
  • E s e c C βˆ’ 1 ( 2 𝑦 ) +

Q12:

Use a trigonometric substitution to evaluate ο„Έ π‘₯ √ 4 π‘₯ βˆ’ 4 9 d 2 , where π‘₯ > 7 2 .

  • A l n C | | | | √ 4 π‘₯ βˆ’ 4 9 7 + 2 π‘₯ 7 | | | | + 2
  • B 1 2 | | | | √ 4 π‘₯ βˆ’ 4 9 7 | | | | + l n C 2
  • C 1 2 | | | | √ 4 π‘₯ βˆ’ 4 9 7 + 4 π‘₯ 7 | | | | + l n C 2
  • D 1 2 | | | | √ 4 π‘₯ βˆ’ 4 9 7 + 2 π‘₯ 7 | | | | + l n C 2
  • E l n C | | | | √ 4 π‘₯ βˆ’ 4 9 7 + 4 π‘₯ 7 | | | | + 2

Q13:

Use an appropriate substitution and then a trigonometric one to evaluate ο„Έ ο„ž 4 βˆ’ π‘₯ π‘₯ π‘₯ d .

  • A 4 ο€Ώ √ π‘₯ 2  + √ 4 βˆ’ π‘₯ + s i n C βˆ’ 1
  • B 4 ο€Ώ √ π‘₯ 2  + √ π‘₯ √ 4 βˆ’ π‘₯ + s i n C
  • C l n C √ π‘₯ 2 +
  • D 4 ο€Ώ √ π‘₯ 2  + √ π‘₯ √ 4 βˆ’ π‘₯ + s i n C βˆ’ 1
  • E l n C √ 4 βˆ’ π‘₯ 2 +

Q14:

Use a trigonometric substitution to evaluate ο„Έ π‘₯ √ 9 βˆ’ π‘₯ 3 2 0 2 d .

  • A 3 2
  • B πœ‹ 3
  • C 1 2
  • D πœ‹ 6
  • E πœ‹ 4

Q15:

Use an appropriate substitution followed by a trigonometric one to evaluate ο„Έ 𝑒 𝑑 ( 𝑒 + 1 ) l n l n 4 3 3 4 3 2 𝑑 2 𝑑 d .

  • A 2 5
  • B βˆ’ 1 5
  • C 7 1 2
  • D 1 5
  • E 7 5

Q16:

Use a trigonometric substitution to evaluate ο„Έ √ 𝑦 βˆ’ 2 5 𝑦 𝑦 2 3 d , where 𝑦 > 5 .

  • A 5 2 ο€» 𝑦 5  βˆ’ 2 5 √ 𝑦 βˆ’ 2 5 2 𝑦 + s e c C βˆ’ 1 2 2
  • B s e c C βˆ’ 1 2 2 ο€» 𝑦 5  βˆ’ 5 √ 𝑦 βˆ’ 2 5 𝑦 +
  • C 1 1 0 ο€» 𝑦 5  + √ 𝑦 βˆ’ 2 5 2 𝑦 + s e c C βˆ’ 1 2 2
  • D 1 1 0 ο€» 𝑦 5  βˆ’ √ 𝑦 βˆ’ 2 5 2 𝑦 + s e c C βˆ’ 1 2 2
  • E 1 1 0 ο€» 𝑦 5  βˆ’ √ 𝑦 βˆ’ 2 5 𝑦 + s e c C βˆ’ 1 2 2

Q17:

Use a trigonometric substitution to evaluate ο„Έ 𝑣 𝑣 ( 1 βˆ’ 𝑣 ) 2 2 d 5 2 .

  • A ο€Ώ 𝑣 √ 1 βˆ’ 𝑣  + 2 3 C
  • B βˆ’ 1 3 ο€Ώ 𝑣 √ 1 βˆ’ 𝑣  + 2 3 C
  • C βˆ’ 1 2 𝑣 + 2 C
  • D 1 3 ο€Ώ 𝑣 √ 1 βˆ’ 𝑣  + 2 3 C
  • E 1 2 𝑣 + 2 C

Q18:

Use a trigonometric substitution to evaluate ο„Έ π‘₯ π‘₯ √ π‘₯ βˆ’ 1 d 2 2 , where π‘₯ > 1 .

  • A l n C | | √ π‘₯ βˆ’ 1 | | βˆ’ π‘₯ βˆ’ 1 2 + 2 2
  • B βˆ’ √ π‘₯ βˆ’ 1 π‘₯ + 2 C
  • C βˆ’ π‘₯ √ π‘₯ βˆ’ 1 + 2 C
  • D √ π‘₯ βˆ’ 1 π‘₯ + 2 C
  • E 1 π‘₯ + C

Q19:

Use a trigonometric substitution to evaluate ο„Έ 4 π‘₯ π‘₯ ( 1 βˆ’ π‘₯ ) √ 3 2 3 2 0 2 2 d .

  • A 4 √ 3 βˆ’ 2 πœ‹ 3
  • B 2 √ 3 βˆ’ 2 πœ‹ 3
  • C 4 √ 3 + 4 πœ‹ 3
  • D 4 √ 3 βˆ’ 4 πœ‹ 3
  • E 4 √ 3 3 + 2 πœ‹ 3

Q20:

Use a trigonometric substitution to evaluate ο„Έ 𝑦 𝑦 √ 𝑦 βˆ’ 1 d 2 .

  • A t a n C βˆ’ 1 | 𝑦 | +
  • B s e c C | 𝑦 | +
  • C l n C 𝑦 + 1 √ 𝑦 βˆ’ 1 βˆ’ 1 𝑦 + 2
  • D s e c C βˆ’ 1 | 𝑦 | +
  • E l n C 𝑦 + 1 √ 𝑦 βˆ’ 1 + 1 𝑦 + 2

Q21:

Use a trigonometric substitution to evaluate ο„Έ 3 π‘₯ √ 1 + 9 π‘₯ d 2 .

  • A l n C | | 1 + 9 π‘₯ + 3 π‘₯ | | + 2
  • B l n C | | √ 1 + 9 π‘₯ | | + 2
  • C l n C | | | | √ 9 + π‘₯ 9 | | | | + 2
  • D l n C | | √ 1 + 9 π‘₯ + 3 π‘₯ | | + 2
  • E l n C | | | | √ 9 + π‘₯ 3 + π‘₯ 3 | | | | + 2

Q22:

Use a trigonometric substitution to evaluate ο„Έ π‘₯ √ π‘₯ + 4 π‘₯ 3 2 d .

  • A ο€» √ π‘₯ + 4  3 βˆ’ 8 √ π‘₯ + 4 + 2 3 2 C
  • B ο€» √ π‘₯ + 4  3 + 4 √ π‘₯ + 4 + 2 3 2 C
  • C ο€» √ π‘₯ + 4  2 + 4 √ π‘₯ + 4 + 2 3 2 C
  • D ο€» √ π‘₯ + 4  3 βˆ’ 4 √ π‘₯ + 4 + 2 3 2 C
  • E √ π‘₯ + 4 2 + 8 √ π‘₯ + 4 + 2 2 C

Q23:

Use a trigonometric substitution to evaluate ο„Έ π‘₯ ( 4 βˆ’ π‘₯ ) 1 0 2 d 3 2 .

  • A 1 √ 3
  • B 1 2 √ 3
  • C 2 √ 3
  • D 1 4 √ 3
  • E 3 4 √ 3

Q24:

Use a trigonometric substitution to evaluate ο„Έ π‘₯ 4 + π‘₯ π‘₯ 2 2 d .

  • A π‘₯ + 2 ο€» π‘₯ 2  + t a n C βˆ’ 1
  • B π‘₯ βˆ’ ο€» π‘₯ 2  + t a n C βˆ’ 1
  • C 1 2 π‘₯ 2 βˆ’ π‘₯ π‘₯ + 4 + t a n C βˆ’ 1 2
  • D π‘₯ βˆ’ 2 ο€» π‘₯ 2  + t a n C βˆ’ 1
  • E 1 2 π‘₯ 2 + π‘₯ π‘₯ + 4 + t a n C βˆ’ 1 2

Q25:

Use an appropriate substitution followed by a trigonometric one to evaluate ο„Έ 2 𝑑 √ 𝑑 + 4 𝑑 √ 𝑑 1 4 1 1 2 d .

  • A 1 3
  • B πœ‹ 1 2
  • C 2 3
  • D πœ‹ 6
  • E πœ‹ 2

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