Pure Mathematics · 3rd secondary · First Term

Calculus - The definite integral and its applications — Methods of integration

About this lesson

This unit covers the definite integral and its applications, beginning with a review of antiderivatives and indefinite integration. It explains the concept of the differential of a function and introduces standard integration formulas. The main techniques taught are integration by substitution and integration by parts, with worked examples and exercises. Applications include finding the equation of a curve given its slope or rate of change, using initial conditions to determine the constant of integration. By the end, students should be able to compute indefinite integrals using substitution and parts, and solve applied problems involving slopes and curve equations.

Main topics in this lesson

  • Antiderivatives and indefinite integration
  • Differentials
  • Standard integration formulas
  • Integration by substitution
  • Integration by parts
  • Applications of indefinite integration

Key terms and vocabulary

  • antiderivative
  • indefinite integral
  • differential
  • integration by substitution
  • integration by parts
  • arbitrary constant
  • standard integrations

Questions and answers on Calculus - The definite integral and its applications — Methods of integration - 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

أوجد قيمة التكامل غير المحدود: ∫ س^2 جتا(س) دس

  • 1 س^2 جا(س) + 2س جتا(س) − 2 جا(س) + ث
  • 2 س^2 جا(س) − 2س جتا(س) + 2 جا(س) + ث
  • 3 −س^2 جا(س) + 2س جتا(س) − 2 جا(س) + ث
  • 4 س^2 جتا(س) + 2 جا(س) + ث
2

أوجد قيمة التكامل غير المحدود: ∫ س^3 ه^(س^2) دس

  • 1 (1/2)(س^2 − 1) ه^(س^2) + ث
  • 2 (1/2)(س^2 + 1) ه^(س^2) + ث
  • 3 س^2 ه^(س^2) + ث
  • 4 (1/4) س^4 ه^(س^2) + ث
3

أوجد قيمة التكامل غير المحدود: ∫ (4 − 3س)^4 دس

  • 1 −(4 − 3س)^5 / 15 + ث
  • 2 (4 − 3س)^5 / 15 + ث
  • 3 −(4 − 3س)^5 / 5 + ث
  • 4 3(4 − 3س)^5 + ث
4

أوجد قيمة التكامل غير المحدود: ∫ س / (س^2 + 1) دس

  • 1 (1/2) لوجارتم (س^2 + 1) + ث
  • 2 لوجارتم (س^2 + 1) + ث
  • 3 2 لوجارتم (س^2 + 1) + ث
  • 4 −1 / (س^2 + 1) + ث
5

أوجد قيمة التكامل غير المحدود: ∫ س جتا(2س) دس

  • 1 (س/2) جا(2س) + (1/4) جتا(2س) + ث
  • 2 (س/2) جا(2س) − (1/4) جتا(2س) + ث
  • 3 −س جا(2س) + (1/2) جتا(2س) + ث
  • 4 س جا(2س) + جتا(2س) + ث
6

أوجد قيمة التكامل غير المحدود: ∫ (2س + 1)^(−3) دس

  • 1 −1 / [4(2س + 1)^2] + ث
  • 2 1 / [4(2س + 1)^2] + ث
  • 3 −1 / [2(2س + 1)^2] + ث
  • 4 −1 / (2س + 1)^2 + ث

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