Exam 1

What is the power rule for integration?

∫x^n dx = (x^(n+1))/(n+1) + C, n ≠ -1.

What is the integral of e^x?

e^x + C.

What is the integral of a^x (where a > 0, a ≠ 1)?

(a^x)/(ln a) + C.

What is the integral of 1/x?

ln|x| + C.

What is the integral of sinh u?

cosh u + C.

What is the integral of cosh u?

sinh u + C.

What is the formula for integration by parts?

∫u dv = uv - ∫v du.

What is the integral of sec^2 x?

tan x + C.

What is the integral of csc^2 x?

-cot x + C.

What is the integral of sec x tan x?

sec x + C.

What is the integral of csc x cot x?

-csc x + C.

What is the method of partial fraction decomposition used for?

To integrate rational functions where the degree of the numerator is less than the degree of the denominator.

What substitution should you use for √(a^2 - x^2)?

Use x = a sin θ and dx = a cos θ dθ.

What substitution should you use for √(a^2 + x^2)?

Use x = a tan θ and dx = a sec² θ dθ.

What substitution should you use for √(x^2 - a^2)?

Use x = a sec θ and dx = a sec θ tan θ dθ.

How do you evaluate an improper integral with an infinite bound?

Convert it into a limit: ∫(a to ∞) f(x) dx = lim(b→∞) ∫(a to b) f(x) dx.

How do you integrate sin^m x cos^n x when one exponent is odd?

If m is odd, save one sin x and use sin^2 x = 1 - cos^2 x. If n is odd, save one cos x and use cos^2 x = 1 - sin^2 x.

How do you integrate sin^m x cos^n x when both exponents are even?

Use power-reduction identities: sin^2 x = (1 - cos 2x)/2, cos^2 x = (1 + cos 2x)/2.

What is a common strategy for solving ∫x e^x dx?

Use integration by parts: ∫x e^x dx = x e^x - ∫e^x dx.

What is a common strategy for solving ∫x ln x dx?

Use integration by parts: ∫x ln x dx = (x^2/2) ln x - ∫(x^2/2)(1/x) dx.

What is a common strategy for solving ∫x sin x dx?

Use integration by parts: ∫x sin x dx = -x cos x + ∫cos x dx.

How do you evaluate an improper integral with an infinite discontinuity?

Convert it into limits at the discontinuity and break into two limits.

Comparison test

Compare larger integral to the one given, if larger converges, then given will also converge. Compare smaller integral to one given, if smaller diverges, then given will also diverge