Techniques of Integration: Substitution I (Day 1)

Flashcards covering substitution basics, known antiderivative formulas, w-substitution, and example problems from Day 1 notes on Techniques of Integration.

What is the main idea of the substitution method in integration?

It 'undoes' the chain rule by letting w = g(x) so that dw = g'(x) dx and rewriting ∫ f(g(x)) g'(x) dx as ∫ f(w) dw.

In substitution, what is dw in terms of x?

dw = g'(x) dx.

For ∫ e^{−cos θ} sin θ dθ, which substitution is natural and what is the result?

Let w = cos θ, then dw = − sin θ dθ, so ∫ e^{−cos θ} sin θ dθ = − ∫ e^{−w} dw = − e^{−w} + C = − e^{−cos θ} + C.

What is ∫ x^n dx for n ≠ −1?

x^{n+1}/(n+1) + C.

What is ∫ e^{a x} dx for a ≠ 0?

(1/a) e^{a x} + C.

What is ∫ a^x dx?

a^x / ln a + C, for a > 0, a ≠ 1.

What is ∫ dx/x?

ln|x| + C.

What is ∫ sin(a x) dx?

− cos(a x) / a + C.

What is ∫ cos(a x) dx?

sin(a x) / a + C.

What is ∫ csc^2(a x) dx?

− cot(a x) / a + C.

What is ∫ sec^2(a x) dx?

tan(a x) / a + C.

What is ∫ csc(x) cot(x) dx?

− csc(x) + C.

What is ∫ sec(x) tan(x) dx?

sec(x) + C.

What is ∫ dx / sqrt(1 - x^2)?

arcsin(x) + C.

What is ∫ dx / (1 + x^2)?

arctan(x) + C.

What is w-substitution?

A technique where you set w equal to the inner function (or inside function) so that the integral becomes easier to evaluate, using dw = inner'(x) dx.

What are the three Integration Methods introduced on Day 1?

1) Known Antiderivative List, 2) Algebraic Simplification, 3) w-Substitution.

How do you decide which method to use for a given integral?

Check if it has a known antiderivative, can be simplified algebraically, requires a w-substitution, or has no closed form.

Example indefinite integral under Group Activity: ∫ (x^3 + 1) dx, what is the antiderivative?

x^4/4 + x + C.

What is the antiderivative of ∫ dx / sqrt(1 - x^2)?

arcsin(x) + C.

What is the antiderivative of ∫ dx / (1 + x^2)?

arctan(x) + C.