Pure Mathematics · 3rd secondary · First Term

Calculus - The definite integral and its applications — Integration of the exponential and logarithmic functions

About this lesson

This unit covers the indefinite integration of exponential and logarithmic functions, along with geometric and physical applications. It begins by reviewing antiderivatives and the concept of indefinite integration, then presents rules for integrating exponential functions like e^x and e^(kx), and logarithmic functions like 1/x. It includes worked examples and exercises for finding integrals of combinations of these functions, often using substitution or recognizing derivatives. The unit also applies integration to find the equation of a curve given its slope (geometric application) and to solve problems involving rates of change (physical application). By the end, a student should be able to compute indefinite integrals of exponential and logarithmic functions, apply integration to find curves from their slopes, and solve basic physical problems involving rates of change.

Main topics in this lesson

  • Integration of exponential functions
  • Integration of logarithmic functions
  • Geometric applications (slope of tangent)
  • Physical applications (rate of change)

Key terms and vocabulary

  • Antiderivative
  • Indefinite integral
  • Arbitrary constant
  • Exponential function
  • Logarithmic function
  • Slope of tangent
  • Rate of change

Questions and answers on Calculus - The definite integral and its applications — Integration of the exponential and logarithmic functions - practice and revision

Multiple choice questions on this lesson in Pure Mathematics for 3rd secondary, First Term, with practice exercises for revision and exam preparation.

1

إذا كان ميل المماس لمنحنى عند أي نقطة عليه يُعطى بالعلاقة dy/dx = e^x + 1/x، وكان المنحنى يمر بالنقطة (1، e)، فأوجد معادلة المنحنى.

  • 1 y = e^x + ln|x|
  • 2 y = e^x + ln|x| + e
  • 3 y = e^x + ln|x| - 1
  • 4 y = e^x - ln|x| + e
2

أوجد ناتج التكامل غير المحدود: ∫ (3 e^(3x) + 2/x) dx

  • 1 e^(3x) + 2 ln|x| + C
  • 2 3 e^(3x) + 2 ln|x| + C
  • 3 9 e^(3x) + 2 ln|x| + C
  • 4 e^(3x) + (2/x^2) + C
3

إذا كان معدل تغير كمية y بالنسبة إلى الزمن t يُعطى بالعلاقة dy/dt = 1/t، وكانت y = 5 عند t = 1، فأوجد قيمة y عند t = e.

  • 1 6
  • 2 5
  • 3 e + 5
  • 4 1
4

أوجد ناتج التكامل غير المحدود: ∫ e^(x+1) dx

  • 1 e^(x+1) + C
  • 2 e^x + C
  • 3 e^(x+1)/(x+1) + C
  • 4 (x+1) e^(x+1) + C
5

أوجد ناتج التكامل غير المحدود: ∫ (2x + 1/x) dx

  • 1 x^2 + ln|x| + C
  • 2 2 + ln|x| + C
  • 3 x^2 - 1/x^2 + C
  • 4 2x^2 + ln|x| + C
6

أوجد ناتج التكامل غير المحدود: ∫ (e^x - 1/x) dx

  • 1 e^x - ln|x| + C
  • 2 e^x + ln|x| + C
  • 3 e^x + 1/x^2 + C
  • 4 e^x - x + C

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