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 6𝑒+2. What is 𝑓(𝑥), given that 𝑓(5)=1ln?

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

Q2:

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

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

Q3:

A curve passes through (0,1) and the tangent at its point (𝑥,𝑦) has slope 6𝑥8𝑥+1. What is the equation of the curve?

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

Q4:

The gradient of the tangent to a curve is 6𝑥+6𝑥sincos. For 𝑥0,𝜋3, the curve has a local minimum value of 4629. 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 8 9 2 1 8 s i n c o s
  • C 𝑦 = 1 6 6 𝑥 + 1 6 6 𝑥 9 5 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 3𝑥𝑥.

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

Q6:

A curve passes through the point (9,4). The slope of its tangent at a point on the curve (𝑥,𝑦) is given by 𝑥(5𝑥+3). Find the equation of the tangent at the point when 𝑥 is equal to 1.

  • A 𝑦 8 𝑥 + 5 4 0 = 0
  • B 𝑥 + 4 , 2 5 5 8 𝑦 = 0
  • C 𝑥 + 4 , 2 5 5 + 8 𝑦 = 0
  • D 𝑦 + 8 𝑥 + 5 2 4 = 0

Q7:

The slope at the point (𝑥,𝑦) on the graph of a function is 5𝑥2𝑥. Find the equation of the curve if it contains the point (𝑒,5𝑒+3).

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

Q8:

The slope at the point (𝑥,𝑦) on the graph of a function is ddsincos𝑦𝑥=4𝜋𝜋𝑥+5𝜋𝜋𝑥. Find the equation of the curve if it contains the point (1,2).

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

Q9:

The second derivative of a curve is 273𝑥+8sin. The curve passes through the point 𝜋6,4𝜋3+𝜋9+6 and the gradient of the tangent at this point is 8+4𝜋3. Find the equation of the curve.

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

Q10:

Find the equation of the curve given the gradient of the tangent is 5𝑥2sin and the curve passes through the origin.

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

Q11:

A curve passes through the points 𝜋4,8 and 3𝜋4,6. Find the equation of the curve given the gradient of the tangent to the curve equals 7(𝑥)csc.

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

Q12:

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

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

Q13:

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

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

Q14:

A curve passes through (1,8) and the normal at its point (𝑥,𝑦) has slope 89𝑥. What is the equation of the curve?

  • A 𝑦 = 8 𝑥 9 2 𝑥 + 9 2
  • B 𝑦 = 1 9 | 8 9 𝑥 | + 8 l n
  • C 𝑦 = 8 𝑥 + 9 2 𝑥 + 2 3 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 45𝑒+4𝑥. What is 𝑓(4𝑒) if we know that 𝑓(𝑒)=9?

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

Q16:

Find the equation of the curve given the gradient of the normal to the curve is 2𝑥2 and the curve passes through the point (1,6).

  • A 𝑦 = 2 𝑥 2 + 6
  • B 𝑦 = 2 2 𝑥 2 + 6
  • C 𝑦 = 1 3 2 𝑥 2 + 6
  • D 𝑦 = 1 4 2 𝑥 2 + 6
  • E 𝑦 = 1 2 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 11𝑥.

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

Q18:

Find the local minimum value of a curve given that its gradient is dd𝑦𝑥=𝑥+3𝑥18 and the local maximum value is 21.

Q19:

The gradient of the tangent to a curve is dd𝑦𝑥=𝑥14𝑥+45 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 5 𝑥 + 9 , 5 4 5 3
  • B 𝑦 = 𝑥 9 𝑥 + 4 5 , 5
  • C 𝑦 = 𝑥 3 7 𝑥 + 4 5 𝑥 2 4 8 3 , 5 3
  • D 𝑦 = 𝑥 3 7 𝑥 + 4 5 𝑥 , 9

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:

The slope at the point (𝑥,𝑦) on the graph of a function is ddsincos𝑦𝑥=𝑥𝑥. Find the equation of the curve if it contains the point 𝜋3,7.

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

Q22:

Find the equation of the curve given the slope of the tangent to the curve at its point (𝑥,𝑦) is cossec𝑥9𝑥 and it passes through the point 𝜋4,22.

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

Q23:

Find the equation of the curve given 𝑦=65𝑥cos 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 2 5 5 𝑥 + 1 3 1 2 5 c o s
  • C 𝑦 = 𝑥 + 6 5 5 𝑥 + 1 9 5 c o s
  • D 𝑦 = 𝑥 + 6 2 5 5 𝑥 + 1 3 1 2 5 c o s

Q24:

A curve passes through (0,1) and the tangent at its point (𝑥,𝑦) has slope 𝑥3𝑥+4. What is the equation of the curve?

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

Q25:

The rate of change in the area 𝐴 of a metallic plate with respect to time due to heating is given by the relation dd𝐴𝑡=0.036𝑡+0.038𝑡, where the area 𝐴 is in square meters and the time 𝑡 is in minutes. Given that 𝐴=67m when 𝑡=8minutes, find, correct to the nearest two decimal places, the area of the plate just before heating.

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