Worksheet: Indefinite Integrals: Exponential and Reciprocal Functions

In this worksheet, we will practice finding the indefinite integral of exponential and reciprocal functions (1/x).

Q1:

Determine 2𝑒𝑥𝑥d.

  • A 2 𝑒 9 1 3 1 8
  • B 2 𝑒 5 2
  • C 2 𝑒 3
  • D 2 𝑒 9 1 1 9

Q2:

Determine 55𝑒𝑛d.

  • A 2 5 𝑒 + C
  • B 5 𝑒 + C
  • C 5 5 𝑒 + C
  • D 5 𝑒 + C

Q3:

Evaluate (𝑥+2𝑒)𝑥d.

  • A 2 𝑒 + 1
  • B 2 𝑒 1 1 + 𝑒 + 2
  • C 2 + 1 1 + 𝑒 + 2 𝑒
  • D 1 + 2 𝑒
  • E 2 + 1 𝑒 + 2 𝑒

Q4:

Determine 8𝑒𝑒+97𝑒𝑥d.

  • A 8 7 𝑒 𝑒 7 + 9 7 𝑒 + C
  • B 4 7 𝑒 𝑒 7 + 9 7 𝑒 + C
  • C 4 7 𝑒 𝑒 7 9 7 𝑒 + C
  • D 1 6 7 𝑒 𝑒 7 9 7 𝑒 + C

Q5:

Given that 𝑓(𝑥)=5𝑒+2𝑥, find 𝑓(𝑥).

  • A 𝑓 ( 𝑥 ) = 5 4 𝑒 + 𝑥 3 + C
  • B 𝑓 ( 𝑥 ) = 5 𝑒 + 2 𝑥 + 𝑥 + C D
  • C 𝑓 ( 𝑥 ) = 5 1 6 𝑒 + 𝑥 2 1 + 𝑥 + C D
  • D 𝑓 ( 𝑥 ) = 5 𝑒 + 𝑥 2 1 + 𝑥 + C D
  • E 𝑓 ( 𝑥 ) = 5 𝑒 + 2 𝑥 + C

Q6:

Determine 3𝑒+8𝑥𝑥d.

  • A 3 𝑒 5 + 1 7 7 5
  • B 3 𝑒 + 3 3
  • C 3 𝑒 + 6 9
  • D 3 𝑒 5 + 3 5 7 5

Q7:

Determine 497𝑥+5𝑥d.

  • A 4 9 𝑥 5 7 | 𝑥 | + l n C
  • B 7 | 7 𝑥 + 5 | + l n C
  • C 7 | 7 𝑥 + 5 | + l n C
  • D 4 9 𝑥 5 4 9 | 𝑥 | + l n C

Q8:

Determine 23𝑥7𝑥lnd.

  • A 2 3 7 | 𝑥 | + l n l n C
  • B 2 3 7 | 𝑥 | + l n l n C
  • C 2 3 7 | 𝑥 | + l n l n C
  • D 2 3 7 | 𝑥 | + l n l n C

Q9:

Determine 4𝜋𝑒𝑥d.

  • A 4 𝜋 3 𝑒 + C
  • B 4 𝜋 𝑒 + C
  • C 4 𝜋 3 𝑒 + C
  • D 4 𝜋 3 𝑒 + C

Q10:

Determine (2𝑥5)𝑥𝑥d.

  • A 4 𝑥 + 2 0 𝑥 + 2 5 | 𝑥 | + l n C
  • B 4 𝑥 + 2 0 𝑥 + 2 5 | 𝑥 | + l n C
  • C 2 𝑥 + 2 0 𝑥 + 2 5 | 𝑥 | + l n C
  • D 2 𝑥 + 2 0 𝑥 + 2 5 | 𝑥 | + l n C

Q11:

Determine 8𝑥+4𝑥𝑥d.

  • A 4 𝑥 5 + 4 | 𝑥 | + l n C
  • B 4 𝑥 5 + 4 | 𝑥 | + l n C
  • C 8 𝑥 + 4 | 𝑥 | + l n C
  • D 8 𝑥 + 4 | 𝑥 | + l n C

Q12:

Determine 9𝑒+52𝑒𝑥d.

  • A 8 1 𝑒 + 4 5 𝑥 + 2 5 4 𝑒 + C
  • B 2 7 4 𝑒 + 4 5 𝑥 2 5 4 8 𝑒 + C
  • C 3 4 𝑒 + 4 5 𝑥 5 4 8 𝑒 + C
  • D 2 7 4 𝑒 + 4 5 𝑥 + 2 5 4 8 𝑒 + C

Q13:

Determine 8𝑥+7𝑒𝑥d.

  • A 8 𝑥 3 𝑒 + 1 + 7 𝑒 + C
  • B 8 𝑥 3 𝑒 + 1 7 8 𝑒 + C
  • C 7 8 𝑒 + 8 3 𝑒 𝑥 + C
  • D 7 8 𝑒 + 8 3 𝑒 𝑥 + C

Q14:

Determine2𝑥d.

  • A 2 9 2 + l n C
  • B 2 2 + l n C
  • C 2 2 + l n C
  • D 2 9 2 + l n C

Q15:

Determine 57𝑒𝑥d.

  • A 1 5 7 𝑒 + C
  • B 5 7 𝑒 + C
  • C 5 7 𝑒 + C
  • D 5 2 1 𝑒 + C

Q16:

Determine 6𝑒𝑦d.

  • A 6 0 𝑒 + C
  • B 0 . 6 𝑒 + C
  • C 6 0 𝑒 + C
  • D 6 𝑒 + C

Q17:

Determine 7𝑥36𝑥𝑥d.

  • A 7 𝑥 1 2 1 2 | 𝑥 | + l n C
  • B 7 𝑥 3 1 2 | 𝑥 | + l n C
  • C 7 𝑥 1 2 1 2 | 𝑥 | + l n C
  • D 7 𝑥 6 1 2 | 𝑥 | + l n C

Q18:

Determine 7𝑒𝑥+2𝑥𝑒𝑥d.

  • A 7 𝑒 | 𝑥 | + 8 𝑥 𝑒 + l n C
  • B 7 𝑒 | 𝑥 | + 2 𝑥 5 𝑒 + l n C
  • C 7 𝑒 | 𝑥 | + 2 𝑥 5 𝑒 + l n C
  • D 7 𝑒 | 𝑥 | + 𝑥 2 𝑒 + l n C

Q19:

Determine 27𝑥𝑥d.

  • A 2 7 | 𝑥 | + l n C
  • B 2 7 | 𝑥 | + l n C
  • C 2 7 | 𝑥 | + l n C
  • D 2 7 | 𝑥 | + l n C

Q20:

Determine the most general antiderivative 𝐹(𝑥) of the function 𝑓, given that 𝑓(𝑥)=52+4𝑥.

  • A 𝐹 ( 𝑥 ) = 5 𝑥 2 + | 𝑥 | + l n C
  • B 𝐹 ( 𝑥 ) = 𝑥 2 + | 𝑥 | + l n C
  • C 𝐹 ( 𝑥 ) = 5 𝑥 + 4 | 𝑥 | + l n C
  • D 𝐹 ( 𝑥 ) = 5 𝑥 2 + 4 | 𝑥 | + l n C
  • E 𝐹 ( 𝑥 ) = 𝑥 2 + 4 | 𝑥 | + l n C

Q21:

Find, if possible, an antiderivative 𝐹 of 𝑓(𝑥)=12𝑥1 that satisfies the conditions 𝐹(0)=1 and 𝐹(1)=1.

  • ANo such antiderivative exists.
  • B 𝐹 ( 𝑥 ) = 1 2 ( 1 2 𝑥 ) + 1 𝑥 < 1 2 , 1 2 ( 2 𝑥 1 ) 1 𝑥 > 1 2 l n f o r l n f o r

Q22:

Determine the most general antiderivative of the function 𝑟(𝜃)=3𝑒+2𝜃𝜃tansec.

  • A 𝑅 ( 𝜃 ) = 3 𝑒 + 2 𝜃 + s e c C
  • B 𝑅 ( 𝜃 ) = 3 𝑒 + 𝜃 𝜃 + t a n s e c C
  • C 𝑅 ( 𝜃 ) = 3 𝑒 𝜃 + 1 + 2 𝜃 + s e c C
  • D 𝑅 ( 𝜃 ) = 𝑒 ( 3 𝜃 + 3 ) + 2 𝜃 + s e c C
  • E 𝑅 ( 𝜃 ) = 3 𝑒 𝜃 + 1 + 𝜃 𝜃 + t a n s e c C

Q23:

Determine 54𝑥𝑥lnd.

  • A 1 4 5 | 𝑥 | + l n l n C
  • B 1 4 ( 𝑥 + 5 ) + l n C
  • C l n l n C 5 | 4 𝑥 | +
  • D 1 4 5 𝑥 + l n C

Q24:

Determine 3𝑥+79𝑥𝑥d.

  • A 9 𝑥 2 + 7 𝑥 3 + 4 9 8 1 | 𝑥 | + l n C
  • B 9 𝑥 2 + 1 4 𝑥 3 + 4 9 8 1 | 𝑥 | + l n C
  • C 9 𝑥 + 1 4 𝑥 3 + 4 9 8 1 | 𝑥 | + l n C
  • D 3 𝑥 2 + 1 4 𝑥 3 + 7 9 | 𝑥 | + l n C

Q25:

Determine 74𝑥6𝑒𝑥d.

  • A 7 4 | 𝑥 | + 3 𝑒 + l n C
  • B 7 4 | 𝑥 | 6 𝑒 + l n C
  • C 7 4 | 𝑥 | + 3 𝑒 + l n C
  • D 7 | 4 𝑥 | + 3 𝑒 + l n C

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