Worksheet: Applications of Indefinite Integration

In this worksheet, we will practice using indefinite integration to express a function given its rate of change.

Q1:

The slope at the point ( 𝑥 , 𝑦 ) on the graph of a function is 6 𝑒 + 2 𝑥 . What is 𝑓 ( 𝑥 ) , given that 𝑓 ( 5 ) = 1 l n ?

  • A 6 𝑒 + 2 𝑥 2 5 + 1 𝑥 l n
  • B 6 𝑒 + 2 𝑥 + 2 5 + 3 1 𝑥 l n
  • C 6 𝑒 + 2 𝑥 + 1 + 2 5 𝑥 l n
  • D 6 𝑒 + 2 𝑥 2 9 2 5 𝑥 l n

Q2:

The area 𝐴 of a lamina is changing at the rate d d 𝐴 𝑡 = 𝑒 0 . 7 𝑡 cm2/s, starting from an area of 60 cm2. Give an exact expression for the area of the lamina after 30 seconds.

  • A 𝑒 + 5 9 2 1 cm2
  • B 1 0 7 𝑒 + 4 1 0 7 2 1 cm2
  • C 1 0 7 𝑒 + 4 1 0 7 2 1 cm2
  • D 1 0 7 𝑒 + 4 3 0 7 2 1 cm2

Q3:

A curve passes through ( 0 , 1 ) and the tangent at its point ( 𝑥 , 𝑦 ) has slope 6 𝑥 8 𝑥 + 1 2 . What is the equation of the curve?

  • A 𝑦 = 1 4 8 𝑥 + 1 + 5 4 2 3 2
  • B 𝑦 = 1 3 2 8 𝑥 + 1 + 3 1 3 2 2 3 2
  • C 𝑦 = 3 1 6 8 𝑥 + 1 + 1 3 1 6 2 3 2
  • D 𝑦 = 1 4 8 𝑥 + 1 + 3 4 2 3 2

Q4:

The gradient of the tangent to a curve is 6 𝑥 + 6 𝑥 s i n c o s . For 𝑥 0 , 𝜋 3 , the curve has a local minimum value of 4 6 2 9 . Find the equation of the curve.

  • A 𝑦 = 1 6 6 𝑥 1 6 6 𝑥 9 5 2 1 8 s i n c o s
  • B 𝑦 = 1 6 6 𝑥 + 1 6 6 𝑥 9 5 2 1 8 s i n c o s
  • C 𝑦 = 1 6 6 𝑥 1 6 6 𝑥 8 9 2 1 8 s i n c o s
  • D 𝑦 = 1 6 6 𝑥 + 1 6 6 𝑥 8 9 2 1 8 s i n c o s

Q5:

Find the equation of a curve which passes through the point ( 0 , 0 ) and, for each point ( 𝑎 , 𝑏 ) on the curve, the slope of the tangent at that point is 3 𝑥 𝑥 5 8 9 .

  • A 𝑦 = 6 2 𝑥 3 6 2 9
  • B 𝑦 = 8 𝑥 1 9 5 7 8
  • C 𝑦 = 6 2 𝑥 9 6 2 9
  • D 𝑦 = 2 7 𝑥 6 2 6 2 9

Q6:

The gradient of the tangent to a curve passing through the point is equal to . Find the equation of the tangent at the point when is equal to 1.

  • A
  • B
  • C
  • D

Q7:

The slope at the point ( 𝑥 , 𝑦 ) on the graph of a function is 5 𝑥 2 𝑥 . Find the equation of the curve if it contains the point ( 𝑒 , 5 𝑒 + 3 ) .

  • A 5 𝑥 2 | 𝑥 | + 1 l n
  • B 5 𝑥 + 5 2 | 𝑥 | l n
  • C 1 0 𝑒 + 5 𝑥 2 | 𝑥 | + 1 l n
  • D 5 𝑥 2 | 𝑥 | + 5 l n

Q8:

The slope at the point ( 𝑥 , 𝑦 ) on the graph of a function is d d s i n c o s 𝑦 𝑥 = 4 𝜋 𝜋 𝑥 + 5 𝜋 𝜋 𝑥 . Find the equation of the curve if it contains the point ( 1 , 2 ) .

  • A 𝑦 = 5 𝜋 𝑥 + 4 𝜋 𝑥 2 s i n c o s
  • B 𝑦 = 5 𝜋 𝜋 𝑥 + 4 𝜋 𝜋 𝑥 + 6 s i n c o s
  • C 𝑦 = 5 𝜋 𝑥 4 𝜋 𝑥 + 6 s i n c o s
  • D 𝑦 = 5 𝜋 𝑥 + 4 𝜋 𝑥 + 6 s i n c o s

Q9:

The second derivative of a curve is 2 7 3 𝑥 + 8 s i n . The curve passes through the point 𝜋 6 , 4 𝜋 3 + 𝜋 9 + 6 2 and the gradient of the tangent at this point is 8 + 4 𝜋 3 . Find the equation of the curve.

  • A 𝑦 = 4 𝑥 8 𝑥 + 9 3 𝑥 + 3 2 s i n
  • B 𝑦 = 4 𝑥 8 𝑥 + 3 3 𝑥 3 2 s i n
  • C 𝑦 = 4 𝑥 8 𝑥 + 9 3 𝑥 3 2 s i n
  • D 𝑦 = 4 𝑥 8 𝑥 + 3 3 𝑥 + 3 2 s i n

Q10:

Find the equation of the curve given the gradient of the tangent is 5 𝑥 2 s i n 2 and the curve passes through the origin.

  • A 𝑦 = 5 𝑥 5 𝑥 s i n
  • B 𝑦 = 5 𝑥 2 c o s 2
  • C 𝑦 = 5 3 𝑥 2 s i n 3
  • D 𝑦 = 5 2 𝑥 5 2 𝑥 s i n

Q11:

A curve passes through the points 𝜋 4 , 8 and 3 𝜋 4 , 6 . Find the equation of the curve given the gradient of the tangent to the curve equals 7 ( 𝑥 ) c s c 2 .

  • A 𝑦 = 7 𝑥 + 1 c s c
  • B 𝑦 = 7 𝑥 + 1 t a n
  • C 𝑦 = 7 𝑥 + 1 c o t
  • D 𝑦 = 7 𝑥 + 1 c o t
  • E 𝑦 = 7 𝑥 + 1 c s c

Q12:

The slope at the point ( 𝑥 , 𝑦 ) on the graph of a function is 3 𝑒 6 𝑥 . What is 𝑓 ( 3 ) , given that 𝑓 ( 5 ) = 9 ?

  • A 9 1 2 𝑒 + 1 2 𝑒 3 3 0
  • B 9 1 8 𝑒 + 1 2 𝑒 1 8 3 0
  • C 9 1 8 𝑒 + 1 2 𝑒 3 3 0
  • D 9 1 2 𝑒 + 1 2 𝑒 1 8 3 0

Q13:

Given that the slope at ( 𝑥 , 𝑦 ) is 3 𝑒 3 𝑥 and 𝑓 ( 0 ) = 3 , determine 𝑓 ( 3 ) .

  • A 4 + 9 𝑒 9
  • B 4 + 3 𝑒 9
  • C 4 + 1 𝑒 3
  • D 4 + 1 𝑒 9

Q14:

A curve passes through ( 1 , 8 ) and the normal at its point ( 𝑥 , 𝑦 ) has slope 8 9 𝑥 . What is the equation of the curve?

  • A 𝑦 = 8 𝑥 9 2 𝑥 + 9 2 2
  • B 𝑦 = 1 9 | 8 9 𝑥 | + 8 l n
  • C 𝑦 = 8 𝑥 + 9 2 𝑥 + 2 3 2 2
  • D 𝑦 = 1 9 | 8 9 𝑥 | + 8 l n

Q15:

The slope at the point ( 𝑥 , 𝑓 ( 𝑥 ) ) on the graph of a function is 4 5 𝑒 + 4 𝑥 . What is 𝑓 ( 4 𝑒 ) if we know that 𝑓 ( 𝑒 ) = 9 ?

  • A 6 4 𝑒 5 1 0 + 4 1 6 𝑒 l n
  • B 4 1 6 𝑒 6 6 5 l n
  • C 1 0 + 1 1 1 𝑒 l n
  • D l n 1 1 𝑒 1 0

Q16:

Find the equation of the curve given the gradient of the normal to the curve is 2 𝑥 2 and the curve passes through the point ( 1 , 6 ) .

  • A 𝑦 = 1 4 2 𝑥 2 + 6
  • B 𝑦 = 1 2 2 𝑥 2 + 6
  • C 𝑦 = 2 2 𝑥 2 + 6
  • D 𝑦 = 2 𝑥 2 + 6
  • E 𝑦 = 1 3 2 𝑥 2 + 6

Q17:

Find the equation of the curve that passes through the point ( 2 , 1 ) given that the gradient of the tangent to the curve is 1 1 𝑥 2 .

  • A 𝑦 = 1 1 3 𝑥 + 3 C
  • B 𝑦 = 1 1 3 𝑥 + 4 7 3 2
  • C 𝑦 = 1 1 𝑥 + 9 3
  • D 𝑦 = 1 1 3 𝑥 8 5 3 3

Q18:

Find the local minimum value of a curve given that its gradient is d d 𝑦 𝑥 = 𝑥 + 3 𝑥 1 8 2 and the local maximum value is 21.

Q19:

The gradient of the tangent to a curve is d d 𝑦 𝑥 = 𝑥 1 4 𝑥 + 4 5 2 where the value of the local maximum is 9. Find the equation of the curve and the value of the local minimum if it exists.

  • A 𝑦 = 𝑥 9 𝑥 + 4 5 2 , 5
  • B 𝑦 = 𝑥 3 7 𝑥 + 4 5 𝑥 3 2 , 9
  • C 𝑦 = 𝑥 5 𝑥 + 9 2 , 5 4 5 3
  • D 𝑦 = 𝑥 3 7 𝑥 + 4 5 𝑥 2 4 8 3 3 2 , 5 3

Q20:

If the rate of change of the sales in a factory is inversely proportional to time in weeks, and the sales of the factory after 2 weeks and 4 weeks are 118 units and 343 units, respectively, determine the sales of the factory after 8 weeks.

Q21:

You are told that 𝑓 ( 𝑥 ) = 1 2 [ 𝑒 + 𝑒 ] 𝑥 𝑥 . If 𝑓 ( 0 ) = 1 and 𝑓 ( 0 ) = 0 , which of the following is equal to 𝑓 ( 𝑥 ) ?

  • A 𝑓 ( 𝑥 )
  • B 𝑓 ( 𝑥 )
  • C 𝑓 ( 𝑥 )
  • D 𝑓 ( 𝑥 )

Q22:

The slope at the point ( 𝑥 , 𝑦 ) on the graph of a function is d 𝑦 d 𝑥 = s i n 𝑥 c o s 𝑥 . Find the equation of the curve if it contains the point 𝜋 3 , 7 .

  • A 𝑦 = 1 2 c o s 2 𝑥 + 2 7 4
  • B 𝑦 = 1 4 c o s 2 𝑥 + 5 7 8
  • C 𝑦 = 1 2 c o s 2 𝑥 + 2 9 4
  • D 𝑦 = 1 4 c o s 2 𝑥 + 5 5 8
  • E 𝑦 = 1 4 c o s 2 𝑥 + 5 5 8

Q23:

Find the equation of the curve given the slope of the tangent to the curve at its point ( 𝑥 , 𝑦 ) is c o s s e c 𝑥 9 𝑥 and it passes through the point 𝜋 4 , 2 2 .

  • A 𝑦 = 𝑥 9 𝑥 + 2 2 + 1 7 s i n t a n
  • B 𝑦 = 𝑥 9 𝑥 1 2 2 s i n t a n
  • C 𝑦 = 𝑥 9 𝑥 9 s i n t a n
  • D 𝑦 = 𝑥 9 𝑥 + 9 s i n t a n

Q24:

Find the equation of the curve given 𝑦 = 6 5 𝑥 c o s and the equation of the tangent to the curve at ( 0 , 5 ) is 𝑦 = 𝑥 + 5 .

  • A 𝑦 = 5 𝑥 6 2 5 5 𝑥 + 1 3 1 2 5 c o s
  • B 𝑦 = 𝑥 + 6 5 5 𝑥 + 1 9 5 c o s
  • C 𝑦 = 𝑥 + 6 2 5 5 𝑥 + 1 3 1 2 5 c o s
  • D 𝑦 = 𝑥 6 2 5 5 𝑥 + 1 3 1 2 5 c o s

Q25:

A curve passes through ( 0 , 1 ) and the tangent at its point ( 𝑥 , 𝑦 ) has slope 4 𝑥 2 𝑥 + 9 2 . What is the equation of the curve?

  • A 𝑦 = 2 3 2 𝑥 + 9 + 1 9 2 3 2
  • B 𝑦 = 1 3 2 𝑥 + 9 8 2 3 2
  • C 𝑦 = 2 2 𝑥 + 9 5 3 2 3 2
  • D 𝑦 = 2 3 2 𝑥 + 9 1 7 2 3 2

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