Worksheet: Integrals Resulting in Inverse Trigonometric Functions

In this worksheet, we will practice evaluating integrals resulting in inverse trigonometric functions, such as ∫ 1 / (1+x²) dx.

Q1:

Find the most general antiderivative 𝐺 ( 𝑣 ) of the function 𝑔 ( 𝑣 ) = 4 𝑣 + 3 5 1 𝑣 c o s .

  • A 𝐺 ( 𝑣 ) = 4 𝑣 + 3 𝑣 5 + s i n s i n C
  • B 𝐺 ( 𝑣 ) = 4 𝑣 + 3 𝑣 5 + s i n c o s C
  • C 𝐺 ( 𝑣 ) = 4 𝑣 3 𝑣 5 + s i n s i n C
  • D 𝐺 ( 𝑣 ) = 4 𝑣 + 3 𝑣 5 + s i n s i n C
  • E 𝐺 ( 𝑣 ) = 4 𝑣 + 3 𝑣 5 + s i n c o s C

Q2:

Evaluate 1 1 + 𝑥 𝑥 d .

  • A 7 𝜋 1 2
  • B 𝜋 1 2
  • C 𝜋 3
  • D 7 𝜋 1 2
  • E 𝜋 1 2

Q3:

Find the most general antiderivative 𝐹 ( 𝑥 ) of the function 𝑓 ( 𝑥 ) = 2 5 𝑥 + 2 9 5 𝑥 + 5 .

  • A 𝐹 ( 𝑥 ) = 5 𝑥 + 4 𝑥 + t a n C
  • B 𝐹 ( 𝑥 ) = 5 𝑥 4 𝑥 + t a n C
  • C 𝐹 ( 𝑥 ) = 5 𝑥 + 4 𝑥 5 + t a n C
  • D 𝐹 ( 𝑥 ) = 5 𝑥 + 4 𝑥 5 + s i n C
  • E 𝐹 ( 𝑥 ) = 5 𝑥 4 𝑥 5 + t a n C

Q4:

What is the antiderivative 𝐹 of 𝑓 ( 𝑥 ) = 5 + 1 + 𝑥 that satisfies 𝐹 ( 1 ) = 0 ?

  • A 𝐹 ( 𝑥 ) = 𝑥 + 𝑥 𝜋 4 + 5 t a n
  • B 𝐹 ( 𝑥 ) = 5 𝑥 + 𝑥 + 1 t a n
  • C 𝐹 ( 𝑥 ) = 5 𝑥 + 𝑥 𝜋 4 + 5 t a n
  • D 𝐹 ( 𝑥 ) = 𝑥 + 𝑥 + 1 t a n
  • E 𝐹 ( 𝑥 ) = 5 𝑥 + 𝑥 + 𝜋 4 + 5 t a n

Q5:

Determine the function 𝑓 ( 𝑡 ) such that 𝑓 ( 𝑡 ) = 2 3 ( 𝑡 + 1 ) , and 𝑓 ( 1 ) = 0 .

  • A 𝑓 ( 𝑡 ) = 2 𝑡 3 + 1 s i n
  • B 𝑓 ( 𝑡 ) = 2 𝑡 3 𝜋 3 s i n
  • C 𝑓 ( 𝑡 ) = 2 𝑡 3 + 𝜋 3 s i n
  • D 𝑓 ( 𝑡 ) = 2 𝑡 3 + 𝜋 6 t a n
  • E 𝑓 ( 𝑡 ) = 2 𝑡 3 𝜋 6 t a n

Q6:

Solve the differential equation d d 𝑦 𝑥 𝑥 + 4 = 3 for 𝑦 given that 𝑦 ( 2 ) = 0 .

  • A 𝑦 = 3 2 𝑥 2 + 3 𝜋 8 t a n
  • B 𝑦 = 3 𝑥 2 3 𝜋 8 t a n
  • C 𝑦 = 3 𝑥 2 + 3 𝜋 8 t a n
  • D 𝑦 = 3 2 𝑥 2 3 𝜋 8 t a n
  • E 𝑦 = 3 2 ( 𝑥 ) 3 𝜋 8 t a n

Q7:

Solve the differential equation 𝑥 𝑦 𝑥 = 𝑥 4 d d for 𝑦 given that 𝑦 ( 2 ) = 0 .

  • A 𝑦 = 𝑥 4 2 𝑥 2 s e c
  • B 𝑦 = 𝑥 4 + 2 𝑥 2 s e c
  • C 𝑦 = 𝑥 4 ( 𝑥 ) s e c
  • D 𝑦 = 2 𝑥 1
  • E 𝑦 = 2 𝑥 + 1

Q8:

Find 𝑥 1 𝑥 + 4 𝑥 + 5 𝑥 d .

  • A 𝑥 + 4 𝑥 + 5 3 𝑥 + 4 𝑥 + 5 + l n C
  • B 𝑥 + 4 𝑥 + 5 3 ( 𝑥 + 2 ) + s i n h C
  • C 𝑥 + 4 𝑥 + 5 3 𝑥 + 4 𝑥 + 5 + l n C
  • D 𝑥 + 4 𝑥 + 5 3 | | 𝑥 + 4 𝑥 + 5 + 𝑥 + 2 | | + l n C
  • E 𝑥 + 4 𝑥 + 5 + 𝑥 + 4 𝑥 + 1 + t a n C

Q9:

Evaluate 𝑥 5 + 4 𝑥 𝑥 d .

  • A s i n C 𝑥 2 3 +
  • B c o s C 𝑥 2 3 +
  • C c o s h C 𝑥 2 3 +
  • D t a n C 𝑥 2 3 +
  • E s i n h C 𝑥 2 3 +

Q10:

Evaluate 𝑥 + 3 5 + 4 𝑥 𝑥 𝑥 d .

  • A 5 𝑥 2 3 + 5 + 4 𝑥 𝑥 + s i n h C
  • B 5 𝑥 2 3 5 + 4 𝑥 𝑥 + s i n C
  • C 5 𝑥 2 3 + 5 + 4 𝑥 𝑥 + s i n C
  • D 5 𝑥 2 3 5 + 4 𝑥 𝑥 + s i n C
  • E 5 𝑥 2 3 5 + 4 𝑥 𝑥 + c o s h C

Q11:

Evaluate 𝑥 𝑥 2 𝑥 8 d .

  • A c o s C 𝑥 1 3 +
  • B s i n h C 𝑥 1 3 +
  • C l n C | | 𝑥 ( 𝑥 1 ) 9 1 | | +
  • D s i n C 𝑥 1 3 +
  • E c o s h C 𝑥 1 3 +

Q12:

Evaluate 𝑥 2 𝑥 𝑥 d .

  • A c o s C ( 𝑥 1 ) +
  • B t a n C ( 𝑥 1 ) +
  • C s i n h C ( 𝑥 1 ) +
  • D s i n C ( 𝑥 1 ) +
  • E c o s h C ( 𝑥 1 ) +

Q13:

Evaluate 𝑥 + 2 𝑥 3 𝑥 + 1 𝑥 d .

  • A 1 2 ( 𝑥 + 1 ) 4 ( 𝑥 + 1 ) 4 2 + t a n C
  • B ( 𝑥 + 1 ) 4 2 ( 𝑥 + 1 ) 4 2 + t a n C
  • C ( 𝑥 + 1 ) 4 ( 𝑥 + 1 ) 4 2 + t a n h C
  • D ( 𝑥 + 1 ) 4 ( 𝑥 + 1 ) 4 2 + t a n C
  • E ( 𝑥 + 1 ) 4 2 ( 𝑥 + 1 ) 4 2 + t a n h C

Q14:

Evaluate ( 𝑥 + 7 ) 𝑥 + 2 𝑥 + 5 𝑥 d .

  • A 1 2 𝑥 + 2 𝑥 + 5 + 3 2 𝑥 + 1 2 + l n t a n C
  • B l n t a n h C 𝑥 + 2 𝑥 + 5 + 3 𝑥 + 1 2 +
  • C 1 2 𝑥 + 2 𝑥 + 5 + 3 𝑥 + 1 2 + l n t a n C
  • D 1 2 𝑥 + 2 𝑥 + 5 + 3 𝑥 + 1 2 + l n t a n h C
  • E l n t a n C 𝑥 + 2 𝑥 + 5 + 3 𝑥 + 1 2 +

Q15:

Evaluate 4 𝑥 + 3 4 𝑥 + 4 𝑥 + 1 7 𝑥 d .

  • A | 2 𝑥 + 1 | + 1 2 2 𝑥 + 1 4 + s i n h C
  • B 4 𝑥 + 4 𝑥 + 1 7 + 2 𝑥 + 1 4 + c o s h C
  • C 4 𝑥 + 4 𝑥 + 1 7 + 2 𝑥 + 1 4 + s i n h C
  • D 4 𝑥 + 4 𝑥 + 1 7 + 1 2 2 𝑥 + 1 4 + s i n h C
  • E 4 𝑥 + 4 𝑥 + 1 7 + 1 2 2 𝑥 + 1 4 + c o s h C

Q16:

Find 𝑥 ( 𝑥 2 𝑥 3 ) d .

  • A 𝑥 1 4 𝑥 2 𝑥 + 5 + C
  • B 𝑥 1 4 𝑥 2 𝑥 3 + C
  • C 1 8 ( 𝑥 1 ) + C
  • D 𝑥 1 4 3 + 2 𝑥 𝑥 + C
  • E 𝑥 1 4 𝑥 2 𝑥 3 + C

Q17:

Evaluate 𝑥 4 𝑥 + 4 𝑥 + 1 7 d .

  • A l n C | | 4 𝑥 + 4 𝑥 + 1 7 + 2 𝑥 + 1 | | +
  • B 1 4 2 𝑥 + 1 4 + c o s h C
  • C 1 4 2 𝑥 + 1 4 + s i n h C
  • D c o s h C 2 𝑥 + 1 4 +
  • E 1 2 2 𝑥 + 1 4 + s i n h C

Q18:

Evaluate 𝑥 𝑥 + 4 𝑥 + 5 d .

  • A c o t h C ( 𝑥 + 2 ) +
  • B t a n C ( 𝑥 + 2 ) +
  • C c o t C ( 𝑥 + 2 ) +
  • D s i n h C ( 𝑥 + 2 ) +
  • E t a n h C ( 𝑥 + 2 ) +

Q19:

Find 𝑥 6 𝑥 𝑥 𝑥 d .

  • A 2 7 2 𝑥 3 3 1 2 ( 2 𝑥 + 6 ) 6 + 6 𝑥 𝑥 + s i n C
  • B 2 7 2 𝑥 3 3 1 2 ( 2 𝑥 + 6 ) 6 𝑥 𝑥 + s i n C
  • C 2 7 2 𝑥 3 3 1 2 ( 𝑥 + 9 ) 6 + 6 𝑥 𝑥 + s i n C
  • D 9 2 𝑥 3 3 1 2 ( 𝑥 + 9 ) 6 𝑥 𝑥 + s i n C
  • E 2 7 2 𝑥 3 3 1 2 ( 𝑥 + 9 ) 6 𝑥 𝑥 + s i n C

Q20:

Evaluate 𝑥 𝑥 𝑥 + 1 d .

  • A 4 3 3 3 ( 2 𝑥 1 ) + t a n C
  • B 2 3 3 3 3 ( 2 𝑥 1 ) + c o t C
  • C 2 3 3 3 3 ( 2 𝑥 1 ) + t a n C
  • D 3 2 3 3 ( 2 𝑥 1 ) + t a n C
  • E 4 3 3 3 ( 2 𝑥 1 ) + c o t C

Q21:

Evaluate 𝑥 + 1 2 𝑥 𝑥 𝑥 d .

  • A 2 ( 𝑥 1 ) 1 ( 𝑥 1 ) + s i n C
  • B 2 ( 𝑥 1 ) 1 ( 𝑥 1 ) + s i n C
  • C 2 ( 𝑥 1 ) 1 ( 𝑥 1 ) + s i n h C
  • D 2 ( 𝑥 1 ) 1 ( 𝑥 1 ) + c o s h C
  • E 1 ( 𝑥 1 ) + C

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