Worksheet: Integrals of Vector-Valued Functions

In this worksheet, we will practice integrating vector-valued functions in 2D and 3D.

Q1:

Calculate the integral ( 𝑡 + 3 ) 𝑡 i j d .

  • A 𝑡 2 + 3 𝑡 + i j C
  • B 𝑡 2 3 𝑡 + i j C
  • C 𝑡 2 + 3 𝑡 + i j C
  • D 𝑡 2 + 3 + i j C
  • E 𝑡 2 3 + i j C

Q2:

Evaluate the integral 𝑡 + 𝑡 + 3 𝑡 𝑡 i j k d .

  • A 2 + 4 + 3 2 i j k
  • B 2 + 4 + 1 6 i j k
  • C i j k 4 3 2
  • D 2 + 1 6 i j k
  • E i j k + 4 + 3 2

Q3:

Calculate the integral 5 𝑡 4 𝑡 4 𝑡 + 8 + 5 𝑡 + 2 𝑡 i j k d .

  • A 5 𝑡 4 𝑡 𝑡 + 8 𝑡 + 5 4 𝑡 + 2 𝑡 + i j k C
  • B 𝑡 𝑡 𝑡 + 8 𝑡 + 5 4 𝑡 + 2 𝑡 + i j k C
  • C 𝑡 + 4 𝑡 5 𝑡 8 𝑡 + 4 5 𝑡 + 2 𝑡 + i j k C
  • D 𝑡 𝑡 𝑡 + 8 𝑡 + 5 4 𝑡 + 2 + i j k C
  • E 4 𝑡 + 𝑡 + 𝑡 + 8 + 5 4 𝑡 + 2 + i j k C

Q4:

Evaluate the integral 2 𝑡 + ( 𝑡 + 1 ) 𝑡 𝑡 i k d .

  • A 2 3 + 4 5 i k
  • B 4 5 + 2 3 i k
  • C 1 3 2 5 + 2 5 6 1 5 i k
  • D 1 2 4 5 + 2 5 6 1 5 i k
  • E 3 2 + 8 i k

Q5:

Evaluate the integral 𝑡 + 1 𝑡 + 1 + 𝑒 𝑡 i j k d .

  • A 3 4 + 2 + 1 + 1 𝑒 i j k l n
  • B 4 3 + 2 + 1 1 𝑒 i j k l n
  • C 3 4 + 2 + 1 1 𝑒 i j k l n
  • D 5 4 + 2 + 1 1 𝑒 i j k l n
  • E 4 3 + 2 + 1 + 1 𝑒 i j k l n

Q6:

Evaluate the integral 𝑡 𝑡 + 𝑡 2 𝑡 + 2 𝑡 2 𝑡 𝑡 s e c t a n c o s s i n c o s d i j k .

  • A 2 1 + 𝜋 8 + 1 4 1 6 i j k
  • B 2 1 + 𝜋 8 1 4 + 1 6 i j k
  • C 2 1 𝜋 8 1 4 + 1 6 i j k
  • D 2 1 + 𝜋 8 + 1 4 + 1 6 i j k
  • E 2 1 𝜋 8 + 1 4 + 1 6 i j k

Q7:

Evaluate the integral 1 𝑡 + 1 + 1 𝑡 + 1 + 𝑡 𝑡 + 1 𝑡 i j k d .

  • A l n l n 2 𝜋 4 1 3 2 i j k
  • B l n l n 2 𝜋 4 + 1 3 2 i j k
  • C l n l n 2 + 𝜋 4 1 2 2 i j k
  • D l n l n 2 + 𝜋 4 + 1 3 2 i j k
  • E l n l n 2 + 𝜋 4 + 1 2 2 i j k

Q8:

Calculate the integral 1 1 + 𝑡 + 1 𝑡 1 + 1 1 + 𝑡 𝑡 i j k d .

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

Q9:

Calculate the integral 4 𝑡 + 3 𝑡 + 4 𝑡 5 + 4 𝑡 5 𝑡 + 3 𝑡 i j k d .

  • A 𝑡 + 𝑡 + 4 3 𝑡 5 𝑡 + 𝑡 5 3 𝑡 + 3 𝑡 + i j k C
  • B 𝑡 + 𝑡 + 4 3 𝑡 5 𝑡 + 𝑡 3 5 𝑡 + 3 𝑡 + i j k C
  • C 𝑡 + 𝑡 + 4 3 𝑡 5 𝑡 + 𝑡 3 5 𝑡 + 3 𝑡 + i j k C
  • D 𝑡 + 𝑡 + 3 4 𝑡 5 + 𝑡 5 3 𝑡 + 3 𝑡 + i j k C
  • E 𝑡 + 𝑡 + 3 4 𝑡 5 𝑡 + 𝑡 5 3 𝑡 + 3 𝑡 + i j k C

Q10:

Calculate the integral ( ( 2 𝑡 ) ) ( 2 𝑡 ) + 1 1 + 𝑡 𝑡 c o s s i n d i j k .

  • A 1 2 ( 2 𝑡 ) + ( 2 𝑡 ) + 𝑡 + s i n c o s c o s i j k C
  • B 1 2 ( 2 𝑡 ) + ( 2 𝑡 ) + ( 𝑡 ) + s i n c o s t a n i j k C
  • C 1 2 ( 2 𝑡 ) + ( 2 𝑡 ) + 𝑡 + s i n c o s s i n i j k C
  • D 1 2 ( 2 𝑡 ) ( 2 𝑡 ) + 𝑡 + c o s s i n c o s i j k C
  • E 1 2 ( 2 𝑡 ) + ( 2 𝑡 ) + 𝑡 + s i n c o s t a n i j k C

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