H: Integration flashcards

Topic 9: H: Integration

Fundamental theorem of calculus

If F′(x) = f(x), then ∫ f(x) dx = F(x) + c and ∫ₐᵇ f(x) dx = F(b) − F(a).

∫ xⁿ dx, ∫ e^(kx) dx, ∫ 1/x dx, ∫ sin kx dx and ∫ cos kx dx

xⁿ⁺¹/(n + 1) + c (n ≠ −1) (1/k)e^(kx) + c ln|x| + c −(1/k) cos kx + c (1/k) sin kx + c

∫ 1/(ax + b) dx and ∫ f′(x)/f(x) dx

(1/a) ln|ax + b| + c ln|f(x)| + c

Use d/dx (sin(x²)) = 2x cos(x²) to find ∫ 2x cos(x²) dx.

sin(x²) + c. A derivative you have found is an integral you now know.

Integration by parts

∫ u (dv/dx) dx = uv − ∫ v (du/dx) dx. Choose u to simplify when differentiated, e.g. ∫ x eˣ dx = x eˣ − eˣ + c.

Integration by substitution

Choose u = g(x), replace dx using du/dx, rewrite everything in u, and change the limits (or substitute back) before evaluating.

Find ∫ 1/(x(x + 1)) dx.

Partial fractions: 1/x − 1/(x + 1). ∫ = ln|x| − ln|x + 1| + c = ln|x/(x + 1)| + c.

Area between a curve and the x-axis when part of it is below the axis

Integrals of parts below the axis are negative. Find the roots, integrate each part separately, and add the sizes.

Area between two curves

∫ₐᵇ (upper − lower) dx, where a and b are the x-coordinates of the intersection points.

Solving a separable differential equation

Write dy/dx = f(x)g(y) as ∫ 1/g(y) dy = ∫ f(x) dx, integrate with one constant, then use the given condition to find it. dy/dx = ky gives y = Ae^(kx).

Integration as the limit of a sum

∫ₐᵇ f(x) dx = lim (δx → 0) Σ f(x)δx: the area is the limit of the sum of thin strips.