complex numbers & eulers formula
Use Euler’s formula to write cos(x) and sin(x) in terms of complex exponentials.
cos(x) = (e^{ix} + e^{-ix})/2 and sin(x) = (e^{ix} - e^{-ix})/2i
State the trig identities for sin^2(theta) and cos^2(theta) used to reduce powers for integration.
sin^2(theta) = (1-cos(2*theta))/2 and cos^2(theta) = (1+cos(2*theta))/2
Which trig substitution should be used for an integral containing sqrt(a^2 - x^2)?
x = a*sin(theta)
Write the partial fraction decomposition form for 1/(x+3)(x-3).
A/(x+3) + B/(x-3)
Does the sequence an = 1/n converge? If so, to what value?
Yes, it converges to 0.
Convert the complex number -3+3i into polar form re^{i*theta} by finding its modulus and argument.
r = 3sqrt{2}, theta = (3*pi)/4;
Polar form: 3sqrt{2}e^{i(3*pi)/4}
Evaluate ∫sin^5(x)cos^2(x) dx.
(1/3)cos^3(x) + (2/5)cos^5(x) - (1/2)cos^7(x) + C
Which trig substitution should be used for an integral containing sqrt(x^2 + a^2)?
x = a*tan(theta)
When is polynomial long division required before performing partial fraction decomposition?
When the degree of the numerator is >= the degree of the denominator.
Evaluate the improper integral (0-3)∫1/sqrt(9-x^2)dx.
pi/2
Use De Moivre's Theorem to compute (1+i)^5.
4(-1-i) = -4-4i
Evaluate ∫tan^3(x)sec^3(x) dx.
(sec^5(x)/5) - (sec^3(x)/3) + C
Evaluate ∫sqrt(1-x^2) dx using the substitution x = sin(theta).
(1/2)(sin^-1(x) + x*sqrt(1-x^2)) + C
Write the partial fraction decomposition form for (x-3)/(x)(x^2 + 1).
A/x + (Bx+C)/(x^2 + 1)
Find the limit of the sequence an = (n^2)/(4n^2 + 3n + 2).
1/4
Evaluate the integral ∫e^{i7x}*cos(6x) dx using complex exponentials. (Leave in exponential form).
(1/2)((1/13i)*e^{13ix} + (1/ix)e^{ix}) + C
Use complex exponentials to prove the identity sin^2(x) = (1/2)(1-cos(2x)).
((e^{ix}-e^{-ix})/2i)^2 =(e^{2ix}-2+e^{-2ix})/(-4) = (1/2) - (1/2)cos(2x)
Set up (but do not evaluate) the integral ∫1/(x^2)*sqrt(x^2+4) dx using x = 2*tan(theta).
∫(2*sec^2(theta))/((2*tan(theta))^2)*sqrt((2*tan(theta))^2 + 4) dtheta
Evaluate ∫1/(x^2-4) dx using partial fractions.
(1/4)ln|x-2| - (1/4)ln|x+2| + C
Determine if (2-inf)∫1/(x)(ln x)^2 dx converges or diverges.
Find the three cube roots of z = -1.
w0 = (1/2) + (sqrt{3}/2)i
w1 = -1
w2 = (1/2) - (sqrt{3}/2)i
Evaluate ∫cos(4x)sin(x) dx and simplify the answer so there are no imaginary numbers.
-(1/10)cos(5x) + (1/6)cos(3x) + C
Evaluate ∫1/(sqrt(16+x^2))^3 dx.
x/(16*sqrt(16 + x^2)) + C
Evaluate ∫(x^4-5x^3+6x^2-18)/(x^3-3x^2) dx. (Hint: Use long division first).
(x^2)/2 - 2x + 2ln(x) - 6/x - 2ln|x-3| + C
Use the Squeeze Theorem to find the limit of the sequence an = (cos^2(3n+2))/(5n-1).
0