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?

  • A6𝑒+2𝑥25+1ln
  • B6𝑒+2𝑥2925ln
  • C6𝑒+2𝑥+1+25ln
  • D6𝑒+2𝑥+25+31ln

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.

  • A107𝑒+4107 cm2
  • B107𝑒+4307 cm2
  • C𝑒+59 cm2
  • D107𝑒+4107 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?

  • A𝑦=1328𝑥+1+3132
  • B𝑦=148𝑥+1+34
  • C𝑦=3168𝑥+1+1316
  • D𝑦=148𝑥+1+54

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.

  • A𝑦=166𝑥166𝑥95218sincos
  • B𝑦=166𝑥166𝑥89218sincos
  • C𝑦=166𝑥+166𝑥95218sincos
  • D𝑦=166𝑥+166𝑥89218sincos

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.

  • A𝑦=27𝑥62
  • B𝑦=8𝑥19
  • C𝑦=62𝑥9
  • D𝑦=62𝑥3

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.

  • A𝑦8𝑥+540=0
  • B𝑥+4,2558𝑦=0
  • C𝑥+4,255+8𝑦=0
  • D𝑦+8𝑥+524=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).

  • A5𝑥+52|𝑥|ln
  • B10𝑒+5𝑥2|𝑥|+1ln
  • C5𝑥2|𝑥|+5ln
  • D5𝑥2|𝑥|+1ln

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).

  • A𝑦=5𝜋𝑥4𝜋𝑥+6sincos
  • B𝑦=5𝜋𝜋𝑥+4𝜋𝜋𝑥+6sincos
  • C𝑦=5𝜋𝑥+4𝜋𝑥+6sincos
  • D𝑦=5𝜋𝑥+4𝜋𝑥2sincos

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.

  • A𝑦=4𝑥8𝑥+33𝑥3sin
  • B𝑦=4𝑥8𝑥+33𝑥+3sin
  • C𝑦=4𝑥8𝑥+93𝑥+3sin
  • D𝑦=4𝑥8𝑥+93𝑥3sin

Q10:

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

  • A𝑦=5𝑥5𝑥sin
  • B𝑦=5𝑥2cos
  • C𝑦=53𝑥2sin
  • D𝑦=52𝑥52𝑥sin

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.

  • A𝑦=7𝑥+1csc
  • B𝑦=7𝑥+1cot
  • C𝑦=7𝑥+1csc
  • D𝑦=7𝑥+1tan
  • E𝑦=7𝑥+1cot

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?

  • A912𝑒+12𝑒
  • B918𝑒+12𝑒
  • C912𝑒+12𝑒
  • D918𝑒+12𝑒

Q13:

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

  • A4+1𝑒
  • B4+3𝑒
  • C4+1𝑒
  • D4+9𝑒

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?

  • A𝑦=8𝑥92𝑥+92
  • B𝑦=19|89𝑥|+8ln
  • C𝑦=8𝑥+92𝑥+232
  • D𝑦=19|89𝑥|+8ln

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?

  • Aln11𝑒10
  • B10+111𝑒ln
  • C416𝑒665ln
  • D64𝑒510+416𝑒ln

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).

  • A𝑦=2𝑥2+6
  • B𝑦=22𝑥2+6
  • C𝑦=132𝑥2+6
  • D𝑦=142𝑥2+6
  • E𝑦=122𝑥2+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.

  • A𝑦=113𝑥+C
  • B𝑦=11𝑥+9
  • C𝑦=113𝑥+473
  • D𝑦=113𝑥853

Q18:

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

  • ALocal minimum value is 34.5.
  • BLocal minimum value is 48.
  • CLocal minimum value is 4.5.
  • DLocal minimum value is 100.5.

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.

  • A𝑦=𝑥5𝑥+9, 5453
  • B𝑦=𝑥9𝑥+45, 5
  • C𝑦=𝑥37𝑥+45𝑥2483, 53
  • D𝑦=𝑥37𝑥+45𝑥, 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.

  • A𝑦=142𝑥+558cos
  • B𝑦=122𝑥+274cos
  • C𝑦=142𝑥+578cos
  • D𝑦=142𝑥+558cos
  • E𝑦=122𝑥+294cos

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.

  • A𝑦=𝑥9𝑥+9sintan
  • B𝑦=𝑥9𝑥122sintan
  • C𝑦=𝑥9𝑥+22+17sintan
  • D𝑦=𝑥9𝑥9sintan

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.

  • A𝑦=5𝑥6255𝑥+13125cos
  • B𝑦=𝑥6255𝑥+13125cos
  • C𝑦=𝑥+655𝑥+195cos
  • D𝑦=𝑥+6255𝑥+13125cos

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?

  • A𝑦=1273𝑥+4+1927
  • B𝑦=193𝑥+4+19
  • C𝑦=293𝑥+479
  • D𝑦=193𝑥+4+179

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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