Title: Math 2250 Integration Review

Q: How is [latex]$\int_a^b f(x) \, dx$[/latex]
related to [latex]$\int_b^a f(x) \, dx$[/latex]?
A: [$]\int_a^b f(x) \, dx = - \int_b^a f(x) \, dx[/$]

Q: If [latex]$k$[/latex] is a constant, [$]\int_a^b k f(x) \, dx = [/$]
A: [$]\int_a^b k f(x) \, dx = k \int_a^b f(x) \, dx[/$]

Q: [$]\int_a^b f(x) + g(x) \, dx = [/$]
A: [$]\int_a^b f(x) + g(x) \, dx = \int_a^b f(x) \, dx + \int_a^b g(x) \, dx.[/$]

Q: The area under the curve f(x) between x=a and x=b is what integral?
A: [$]\text{Area} = \int_a^b f(x) \, dx[/$]

Q: The average value of f(x) over (a,b) is what integral?
A: [$]\text{Average} = \frac{1}{b-a} \int_a^b f(x) \, dx[/$]

Q: If [latex]$F(x) = \int_a^x f(t) \, dt$[/latex]
then [latex]$F'(x) = $[/latex]
A: [$]F'(x) = f(x)[/$]

Q: If [latex]F'(x) = f(x)[/latex], then [$]\int_a^b f(x) \, dx = [/$]
A: [$]\int_a^b f(x) \, dx = F(b) - F(a)[/$]

Q: [$]\int x^n \, dx = [/$]
A: [$]\int x^n \, dx = \frac{x^{n+1}}{n+1} + C[/$]

Q: [$]\int \sin x \, dx = [/$]
A: [$]\int \sin x \, dx = - \cos x + C[/$]

Q: [$]\int e^x \, dx =[/$]
A: [$]\int e^x \, dx = e^x + C[/$]

Q: [$]\int \ln x \, dx = [/$]
A: [$]\int \ln x \, dx = x \ln x - x + C[/$]

Q: [$]\int \frac{1}{x} \, dx = [/$]
A: [$]\int \frac{1}{x} \, dx = \ln x + C[/$]

Q: [$]\int \frac{1}{x \sqrt{x^2 - 1}} =[/$]
A: [$]\int \frac{1}{x \sqrt{x^2 - 1}} = \operatorname{arcsec} x + C[/$]

Q: What are the steps for integrating
[$]\int f(g(x)) g'(x) dx[/$]
by [latex]$u$-[/latex]substitution?
A: 1. Write [latex]$u = g(x)$[/latex].
2. Solve [latex]$du = g'(x) dx$[/latex] for [latex]$dx$[/latex].
3. Substitute [latex]$u$[/latex] for [latex]$g(x)$[/latex] and [latex]$du/g'(x)$[/latex] for [latex]$dx$[/latex].
4. Eliminate all [latex]$x$[/latex]'s, leaving only [latex]$u$[/latex] and [latex]$du$[/latex].
5. Integrate with respect to [latex]$u$[/latex].

Q: [$]\int \cos x \, dx = [/$]
A: [$]\int \cos x \, dx = \sin x + C[/$]

Q: [$]\int \tan x \, dx = [/$]
A: [$]\int \tan x \, dx = -\ln \cos x + C[/$]