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Lesson Worksheet: Indefinite Integrals and Initial Value Problems Mathematics • Higher Education

In this worksheet, we will practice using integration to find particular solutions to initial value problems involving differential equations of the form y’ = f(x).


Find the solution of the differential equation ddsec𝑒𝑑=𝑑+π‘‘π‘’οŠ¨ that satisfies the initial condition 𝑒(0)=βˆ’3.

  • A𝑒=𝑑+2𝑑+9tan
  • B𝑒=𝑑+2π‘‘βˆ’9tan
  • C𝑒=π‘‘βˆ’2𝑑+9tan
  • D𝑒=𝑑+2π‘‘οŠ¨οŠ¨tan
  • E𝑒=𝑑+𝑑+9tan


Find the solution of the differential equation 𝑦′π‘₯=π‘Ž+𝑦tan, where 0<π‘₯<πœ‹2, that satisfies the initial condition π‘¦ο€»πœ‹3=π‘Ž.

  • A𝑦=4π‘Žπ‘₯βˆ’π‘Žsin
  • B𝑦=4π‘Žπ‘₯βˆ’π‘Žcos
  • C𝑦=π‘Žβˆ’4π‘Žβˆš3π‘₯sin
  • D𝑦=4π‘Žβˆš3π‘₯βˆ’π‘Žsin
  • E𝑦=π‘Žβˆ’4π‘Žπ‘₯cos


Solve the differential equation dd𝑦π‘₯√π‘₯βˆ’9=1 for 𝑦 given that 𝑦(5)=3ln.

  • A𝑦=ο€»π‘₯+√π‘₯βˆ’9+3lnln
  • B𝑦=ο€»π‘₯+√π‘₯βˆ’9ο‡βˆ’3lnln
  • C𝑦=βˆ’13√π‘₯βˆ’9π‘₯+1345lnln
  • D𝑦=13√π‘₯βˆ’9π‘₯+1345lnln
  • E𝑦=ο€»π‘₯+√π‘₯βˆ’9ο‡βˆ’23lnln


You are told that 𝑓′′(π‘₯)=12[𝑒+𝑒]ο—οŠ±ο—. If 𝑓(0)=1 and 𝑓′(0)=0, which of the following is equal to 𝑓(π‘₯)?

  • A𝑓′′(π‘₯)
  • B𝑓′(π‘₯)
  • Cβˆ’π‘“β€²(π‘₯)
  • Dβˆ’π‘“β€²β€²(π‘₯)


Find the function whose first derivative is 27π‘₯βˆ’83π‘₯βˆ’2 given the value of the function equals βˆ’1 when π‘₯ equals βˆ’1.

  • A𝑓(π‘₯)=3π‘₯+3π‘₯+4π‘₯+3
  • B𝑓(π‘₯)=9π‘₯+6π‘₯+4π‘₯+6
  • C𝑓(π‘₯)=27π‘₯4βˆ’8π‘₯βˆ’634οŠͺ
  • D𝑓(π‘₯)=3π‘₯βˆ’2π‘₯βˆ’6


Suppose that ddcsc𝑦π‘₯=8π‘₯ and 𝑦=βˆ’12 when π‘₯=πœ‹3. Find 𝑦 in terms of π‘₯.

  • Aβˆ’8π‘₯+12cot
  • Bβˆ’8π‘₯βˆ’8√33+12cot
  • Cβˆ’8π‘₯βˆ’12+8√33cot
  • D8π‘₯βˆ’12+8√33cot
  • Eβˆ’8π‘₯βˆ’12cot


Find the function𝑓, if 𝑓′(𝑑)=2𝑑(𝑑+4𝑑)sectansec, when βˆ’πœ‹2<𝑑<πœ‹2 and π‘“ο€»βˆ’πœ‹3=βˆ’2.

  • A𝑓(𝑑)=8𝑑+2π‘‘βˆ’6+8√3tansec
  • B𝑓(𝑑)=8𝑑+2π‘‘βˆ’8√3+2tansec
  • C𝑓(𝑑)=4𝑑+π‘‘βˆ’6+8√3tansec
  • D𝑓(𝑑)=2𝑑+8π‘‘βˆ’8√3+2tansec
  • E𝑓(𝑑)=2𝑑+8π‘‘βˆ’6+8√3tansec


Determine the function 𝑓 satisfying 𝑓′′(πœƒ)=2πœƒ+3πœƒsincos, 𝑓(0)=5, and 𝑓′(0)=1.

  • A𝑓(πœƒ)=3πœƒβˆ’2πœƒ+3sincos
  • B𝑓(πœƒ)=βˆ’2πœƒβˆ’3πœƒβˆ’1sincos
  • C𝑓(πœƒ)=3πœƒβˆ’2πœƒβˆ’3πœƒβˆ’2sincos
  • D𝑓(πœƒ)=3πœƒβˆ’2πœƒβˆ’3πœƒ+8sincos
  • E𝑓(πœƒ)=βˆ’πœƒβˆ’2πœƒβˆ’3πœƒ+2sincos


Determine the function 𝑓 if 𝑓′′′(π‘₯)=4π‘₯cos, 𝑓(0)=βˆ’1, 𝑓′(0)=4, and 𝑓′′(0)=βˆ’4.

  • A𝑓(π‘₯)=βˆ’4π‘₯+8π‘₯βˆ’4π‘₯βˆ’1sin
  • B𝑓(π‘₯)=2π‘₯+8π‘₯+π‘₯+1sin
  • C𝑓(π‘₯)=4π‘₯βˆ’8π‘₯βˆ’4π‘₯βˆ’1sin
  • D𝑓(π‘₯)=βˆ’2π‘₯+8π‘₯βˆ’4π‘₯βˆ’1sin
  • E𝑓(π‘₯)=βˆ’4π‘₯βˆ’8π‘₯βˆ’4π‘₯+1sin


Suppose that ddsincos𝑦π‘₯=βˆ’92π‘₯βˆ’35π‘₯ and 𝑦=7 when π‘₯=πœ‹6. Find 𝑦 in terms of π‘₯.

  • A𝑦=βˆ’355π‘₯+922π‘₯+10120sincos
  • B𝑦=βˆ’92π‘₯βˆ’35π‘₯+13sincos
  • C𝑦=925π‘₯+352π‘₯+8920sincos
  • D𝑦=βˆ’352π‘₯βˆ’925π‘₯+8920sincos
  • E𝑦=βˆ’922π‘₯βˆ’355π‘₯+8920sincos

This lesson includes 13 additional questions and 153 additional question variations for subscribers.

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