The portal has been deactivated. Please contact your portal admin.

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).

Q1:

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

Q2:

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

Q3:

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

Q4:

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‘₯)

Q5:

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

Q6:

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

Q7:

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

Q8:

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

Q9:

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

Q10:

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.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.