Lesson Worksheet: Applications of Indefinite Integration Mathematics

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 that passes through the point (0,0) and is such that for each point (𝑥,𝑦) 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

This lesson includes 31 additional questions and 357 additional question variations for subscribers.

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