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Topic 4: Introduction to integration

Anti-differentiation

9 hours

• Recognise anti-differentiation as the reverse of differentiation. 1 interactive
• Use the notation ∫ 𝑓(𝑥)𝑑𝑥 for anti-derivatives or indefinite integrals.
• Use the formula ∫ 𝑥𝑛 𝑑𝑥 = 𝑥𝑛+1 𝑛+1 + 𝑐 for 𝑛 ≠ − 1.
• Use the formula ∫ 𝑒𝑥 𝑑𝑥 = 𝑒𝑥 + 𝑐.
• Use the formula ∫ 1 𝑥 𝑑𝑥 = ln(𝑥)+ 𝑐, for 𝑥 > 0.
• Use the formulas ∫ sin(𝑥)𝑑𝑥= − cos(𝑥) + 𝑐 and ∫ cos(𝑥)𝑑𝑥 = sin(𝑥)+ 𝑐.
• Understand and use the formulas ∫(𝑓(𝑥)+ 𝑔(𝑥))𝑑𝑥 = ∫ 𝑓(𝑥)𝑑𝑥+ ∫ 𝑔(𝑥)𝑑𝑥 and ∫ 𝑘 𝑓(𝑥)𝑑𝑥 = 𝑘 ∫ 𝑓(𝑥)𝑑𝑥.
• Determine indefinite integrals of the form ∫ 𝑓(𝑎𝑥 + 𝑏)𝑑𝑥.
• Determine 𝑓(𝑥) given 𝑓′(𝑥) and an initial condition 𝑓(𝑎) = 𝑏. 1 interactive
• Determine displacement given velocity and the initial value of displacement. 1 interactive
• Determine displacement given acceleration and initial values of displacement and velocity.
• Model and solve problems that involve indefinite integrals, with and without technology.