MAT137Y1 – LEC0501
Calculus!
Part 2:
Antiderivatives and indefinite integrals
January 16 th , 2019
Jean-Baptiste Campesato MAT137Y1 – LEC0501 – Calculus! – Jan 16, 2019 1
Initial Value Problem
Find a function 𝑓 ∶ ℝ → ℝ such that
• For every 𝑥 ∈ ℝ, 𝑓 ″ (𝑥) = sin 𝑥 + 𝑥 2 ,
• 𝑓 ′ (0) = 5,
• 𝑓 (0) = 7.
Jean-Baptiste Campesato MAT137Y1 – LEC0501 – Calculus! – Jan 16, 2019 2
Towards FTC
−1 1 2 3 4 5
−2
−1 1 2
𝑂
𝑦 = 𝑓 (𝑥)
Compute:
1 ∫
1 0
𝑓 (𝑡)𝑑𝑡
2 ∫
2 0
𝑓 (𝑡)𝑑𝑡
3 ∫
3 0
𝑓 (𝑡)𝑑𝑡
4 ∫
4 0
𝑓 (𝑡)𝑑𝑡
5 ∫
5 0
𝑓 (𝑡)𝑑𝑡
Jean-Baptiste Campesato MAT137Y1 – LEC0501 – Calculus! – Jan 16, 2019 3
Towards FTC (continued)
−1 1 2 3 4 5
−2
−1 1 2
𝑂
𝑦 = 𝑓 (𝑥)
Call 𝐹 (𝑥) =
∫
𝑥 0
𝑓 (𝑡)𝑑𝑡. This is a new function.
• Sketch the graph of 𝑦 = 𝐹 (𝑥).
• Using the graph you just sketched, sketch the graph of 𝑦 = 𝐹 ′ (𝑥).
Jean-Baptiste Campesato MAT137Y1 – LEC0501 – Calculus! – Jan 16, 2019 4
Trig-exp antiderivatives
1 Compute 𝑑
𝑑𝑥 [ 𝑒 𝑥 sin 𝑥] , 𝑑
𝑑𝑥 [ 𝑒 𝑥 cos 𝑥] .
2 Use the previous answer to Compute
∫ 𝑒 𝑥 sin 𝑥 𝑑𝑥,
∫ 𝑒 𝑥 cos 𝑥 𝑑𝑥.
Jean-Baptiste Campesato MAT137Y1 – LEC0501 – Calculus! – Jan 16, 2019 5
A harder antiderivative
1 Compute 𝑑
𝑑𝑥 [arctan 𝑥] , 𝑑 𝑑𝑥 [
𝑥 1 + 𝑥 2 ] .
2 Use the previous answer to compute
∫
1 (1 + 𝑥 2 ) 2
𝑑𝑥
Jean-Baptiste Campesato MAT137Y1 – LEC0501 – Calculus! – Jan 16, 2019 6
