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 ) 𝑥 𝑥 d .

  • A 1 3 𝑥 − 1 5 𝑥 + C
  • B − 1 5 √ 1 − 𝑥 𝑥 + C
  • C 1 3 𝑥 + 1 5 𝑥 + C
  • D 1 5 √ 1 − 𝑥 𝑥 + C
  • E − 1 3 𝑥 − 1 5 𝑥 + C

Q2:

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

  • A − 1 3 𝑥 √ 𝑥 − 1 + C
  • B 𝑥 √ 𝑥 − 1 + C
  • C 1 4 𝑥 √ 𝑥 − 1 + C
  • D − 1 4 𝑥 √ 𝑥 − 1 + C
  • E 1 3 𝑥 √ 𝑥 − 1 + C

Q3:

Use a trigonometric substitution to evaluate √ 𝑦 − 4 9 𝑦 𝑦 d , where 𝑦 > 7 .

  • A 7 𝑦 + C
  • B − 7 𝑦 + C
  • C √ 𝑦 − 4 9 − 7 𝑦 7 + s e c C
  • D 7 √ 𝑦 − 4 9 − 𝑦 7 + s e c C
  • E s e c C 𝑦 7 − √ 𝑦 − 4 9 +

Q4:

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

  • A √ 2 4
  • B 𝜋 8
  • C 𝜋 4
  • D 𝜋 2
  • E √ 2 2

Q5:

Use a trigonometric substitution to evaluate √ 1 − ( 𝑥 ) 𝑥 𝑥 𝑥 l n l n d .

  • A √ 1 − ( 𝑥 ) + | | | 𝑥 1 + √ 1 − 𝑥 | | | + l n l n l n l n C
  • B √ 1 − ( 𝑥 ) − | | | 𝑥 1 + √ 1 − ( 𝑥 ) | | | + l n l n l n l n C
  • C √ 1 − 𝑥 + | | | 𝑥 1 + √ 1 − ( 𝑥 ) | | | + l n l n l n l n C
  • D √ 1 + ( 𝑥 ) + | | | 𝑥 1 + √ 1 − ( 𝑥 ) | | | + l n l n l n l n C
  • E √ 1 − ( 𝑥 ) + | | | 𝑥 1 + √ 1 − ( 𝑥 ) | | | + l n l n l n l n C

Q6:

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

  • A − 𝑥 √ 𝑥 − 1 + C
  • B − 1 √ 𝑥 − 1 + C
  • C 𝑥 √ 𝑥 − 1 + C
  • D 𝑥 √ 𝑥 − 1 + C
  • E − 𝑥 √ 𝑥 − 1 + C

Q7:

Use a trigonometric substitution to evaluate 𝑥 𝑥 √ 1 + 𝑥 d .

  • A 2 𝑥 + √ 1 + 𝑥 + l n C
  • B 2 𝑥 + √ 1 + 𝑥 + l n C
  • C 1 2 𝑥 − √ 1 + 𝑥 + l n C
  • D 1 2 𝑥 + √ 1 + 𝑥 + l n C
  • E 1 2 𝑥 + √ 1 + 𝑥 + l n C

Q8:

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

  • A l n C | | | | √ 9 + 𝑥 9 | | | | +
  • B l n C | | | | √ 9 + 𝑥 3 | | | | +
  • C l n C | | | | √ 9 + 𝑥 9 + 𝑥 9 | | | | +
  • D l n C | | | | √ 9 + 𝑥 3 + 𝑥 3 | | | | +
  • E l n C | | | 9 + 𝑥 3 + 𝑥 3 | | | +

Q9:

Use a trigonometric substitution to evaluate 𝑥 4 + 𝑥 d .

  • A 1 2
  • B2
  • C 𝜋 8
  • D 𝜋 4
  • E 𝜋 4

Q10:

Use trigonometric substitution to evaluate 𝑥 √ 1 − 𝑥 d .

  • A c o s C 𝑥 +
  • B s i n C 𝑥 +
  • C c o s C 𝑥 +
  • D s i n C 𝑥 +
  • E t a n C 𝑥 +

Q11:

Use a trigonometric substitution to evaluate 𝑦 1 + 𝑦 d .

  • A s e c C ( 2 𝑦 ) +
  • B 1 2 ( 𝑦 ) + 𝑦 2 + 2 𝑦 + t a n C
  • C t a n C ( 𝑦 ) +
  • D s e c C ( 𝑦 ) +
  • E 1 2 ( 𝑦 ) − 𝑦 2 + 2 𝑦 + t a n C

Q12:

Use a trigonometric substitution to evaluate 𝑥 √ 4 𝑥 − 4 9 d , where 𝑥 > 7 2 .

  • A 1 2 | | | | √ 4 𝑥 − 4 9 7 + 4 𝑥 7 | | | | + l n C
  • B 1 2 | | | | √ 4 𝑥 − 4 9 7 + 2 𝑥 7 | | | | + l n C
  • C l n C | | | | √ 4 𝑥 − 4 9 7 + 2 𝑥 7 | | | | +
  • D 1 2 | | | | √ 4 𝑥 − 4 9 7 | | | | + l n C
  • E l n C | | | | √ 4 𝑥 − 4 9 7 + 4 𝑥 7 | | | | +

Q13:

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

  • A 4 √ 𝑥 2 + √ 𝑥 √ 4 − 𝑥 + s i n C
  • B 4 √ 𝑥 2 + √ 𝑥 √ 4 − 𝑥 + s i n C
  • C l n C √ 4 − 𝑥 2 +
  • D 4 √ 𝑥 2 + √ 4 − 𝑥 + s i n C
  • E l n C √ 𝑥 2 +

Q14:

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

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

Q15:

Use an appropriate substitution followed by a trigonometric one to evaluate 𝑒 𝑡 ( 𝑒 + 1 ) l n l n d .

  • A − 1 5
  • B 7 5
  • C 1 5
  • D 7 1 2
  • E 2 5

Q16:

Use a trigonometric substitution to evaluate √ 𝑦 − 2 5 𝑦 𝑦 d , where 𝑦 > 5 .

  • A 5 2 𝑦 5 − 2 5 √ 𝑦 − 2 5 2 𝑦 + s e c C
  • B s e c C 𝑦 5 − 5 √ 𝑦 − 2 5 𝑦 +
  • C 1 1 0 𝑦 5 − √ 𝑦 − 2 5 𝑦 + s e c C
  • D 1 1 0 𝑦 5 − √ 𝑦 − 2 5 2 𝑦 + s e c C
  • E 1 1 0 𝑦 5 + √ 𝑦 − 2 5 2 𝑦 + s e c C

Q17:

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

  • A 𝑣 √ 1 − 𝑣 + C
  • B − 1 2 𝑣 + C
  • C 1 2 𝑣 + C
  • D − 1 3 𝑣 √ 1 − 𝑣 + C
  • E 1 3 𝑣 √ 1 − 𝑣 + C

Q18:

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

  • A 1 𝑥 + C
  • B − 𝑥 √ 𝑥 − 1 + C
  • C √ 𝑥 − 1 𝑥 + C
  • D − √ 𝑥 − 1 𝑥 + C
  • E l n C | | √ 𝑥 − 1 | | − 𝑥 − 1 2 +

Q19:

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

  • A 2 √ 3 − 2 𝜋 3
  • B 4 √ 3 3 + 2 𝜋 3
  • C 4 √ 3 − 2 𝜋 3
  • D 4 √ 3 + 4 𝜋 3
  • E 4 √ 3 − 4 𝜋 3

Q20:

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

  • A s e c C | 𝑦 | +
  • B t a n C | 𝑦 | +
  • C l n C 𝑦 + 1 √ 𝑦 − 1 − 1 𝑦 +
  • D l n C 𝑦 + 1 √ 𝑦 − 1 + 1 𝑦 +
  • E s e c C | 𝑦 | +

Q21:

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

  • A l n C | | | | √ 9 + 𝑥 3 + 𝑥 3 | | | | +
  • B l n C | | 1 + 9 𝑥 + 3 𝑥 | | +
  • C l n C | | √ 1 + 9 𝑥 | | +
  • D l n C | | √ 1 + 9 𝑥 + 3 𝑥 | | +
  • E l n C | | | | √ 9 + 𝑥 9 | | | | +

Q22:

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

  • A √ 𝑥 + 4 2 + 4 √ 𝑥 + 4 + C
  • B √ 𝑥 + 4 3 + 4 √ 𝑥 + 4 + C
  • C √ 𝑥 + 4 3 − 8 √ 𝑥 + 4 + C
  • D √ 𝑥 + 4 2 + 8 √ 𝑥 + 4 + C
  • E √ 𝑥 + 4 3 − 4 √ 𝑥 + 4 + C

Q23:

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

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

Q24:

Use a trigonometric substitution to evaluate 𝑥 4 + 𝑥 𝑥 d .

  • A 𝑥 + 2 𝑥 2 + t a n C
  • B 1 2 𝑥 2 + 𝑥 𝑥 + 4 + t a n C
  • C 𝑥 − 2 𝑥 2 + t a n C
  • D 𝑥 − 𝑥 2 + t a n C
  • E 1 2 𝑥 2 − 𝑥 𝑥 + 4 + t a n C

Q25:

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

  • A 𝜋 1 2
  • B 𝜋 2
  • C 𝜋 6
  • D 2 3
  • E 1 3

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