Rule/Identity Memorization Flashcards
What is the derivative of u * v? (Product Rule)
uv’ + vu’
What is the derivative of u / v? (Quotient Rule)
vu’ - uv’ / v^2
What is the derivative of u^n? (Power Rule)
nu^(n-1)*u’
What is the derivative of |u|?
u / |u| * u’ where u != 0
What is the derivative of ln u?
u’ / u
What is the derivative of e^u?
e^u * u’
What is the derivative of log_a_u?
u’ / (ln a) * u
What is the derivative of a^u?
(ln a) * a^u * u’
What is the derivative of sin u?
cos u * u’
What is the derivative of cos u?
-(sin u) * u’
What is the derivative of tan u?
sec^2 u * u’
What is the derivative of cot u?
-(csc^2 u) * u’
What is the derivative of sec u?
(sec u tan u) * u’
What is the derivative of csc u?
-(csc u cot u) * u’
What is the derivative of arcsin u?
u’ / sqrt(1 - u^2)
What is the derivative of arccos u?
-u’ / sqrt(1 - u^2)
What is the derivative of arctan u?
u’ / 1 + u^2
What is the derivative of arccot u?
-u’ / 1 + u^2
What is the derivative of arcsec u?
u’ / |u| * sqrt(u^2 - 1)
What is the derivative of arccsc u?
-u’ / |u| * sqrt(u^2 -1)
What is the derivative of sinh u?
(cosh u) * u’
What is the derivative of cosh u?
(sinh u) * u’
What is the derivative of tanh u?
(sech^2 u) * u’
What is the derivative of coth u?
-(csch^2 u) * u’
What is the derivative of sech u?
-(sech u tanh u) * u’
What is the derivative of csch u?
-(csch u coth u) * u’
What is the integral of a^u du?
(1 / ln a) * a^u + C
What is the integral of e^u du?
e^u + C
What is the integral of sin u du?
-cos u + C
What is the integral of cos u du?
sin u + C
What is the integral of tan u du?
-ln|cos u| + C
What is the integral of cot u du?
ln |sin u| + C
What is the integral of sec u du?
ln |sec u + tan u| + C
What is the integral of csc u du?
-ln |csc u + cot u| + C
What is the integral of sec^2 u du?
tan u + C
What is the integral of csc^2 u du?
-cot u + C
What is the integral of sec u tan u du?
sec u + C
What is the integral of csc u cot u du?
-csc u + C
What is the integral of du / sqrt(a^2 - u^2)?
arcsin u/a + C
What is the integral of du / a^2 + u^2?
1/a arctan u/a + C
What is the integral of du / u * sqrt(u^2 - a^2)?
1/a arcsec |u|/a + C
What is sin 2u?
2 sin u cos u
What is cos 2u?
cos^2 u - sin^2 u = 2 cos^2 u - 1 = 1 - 2 sin^2 u
What is tan 2u?
2 tan u / 1 - tan^2 u
What is the sin/cos Pythagorean Identity?
sin^2 x + cos^2 x = 1
What is the tan/sec Pythagorean Identity?
1 + tan^2 x = sec^2 x
What is the cot/csc Pythagorean Identity?
1 + cot^2 x = csc^2 x
What are the methods used to find ways to integrate?
Expanding, Subsitution, Rewriting as the sum of two quotients (x + a / b) = (x/b) + (a/b), Completing the Square, Dividing improper rational function (x^2 / x^2 + 1), Adding and subtracting terms in the numerator (+1 - 1), Trigonometric Identities, Pythagorean Conjugate
What is sinh x?
e^x - e^-x / 2
What is cosh x?
e^x + e^-x / 2
What is tanh x?
sinh x / cosh x
What is the cosh/sinh hyperbolic identity?
cosh^2 x - sinh^2 x = 1
What is the tanh/sech hyperbolic identity?
tanh^2 x + sech^2 x = 1
What is the coth/csch hyperbolic identity?
coth^2 x - csch^2 x = 1
What is sinh 2x?
2 sinhx cosh x
What is cosh 2x?
cosh^2 x + sinh^2 x
What is the derivative of sinh u?
cosh u * u’
What is the derivative of cosh u?
sinh u * u’
What is the derivative of tanh u?
sech^2 u * u’
What is the derivative of coth u?
-csch^2 u * u’
What is the derivative of sech u?
-(sech u tanh u) * u’
What is the derivative of csch u?
-(csch u coth u) * u’
What is the integral of cosh u du?
sinh u + C
What is the integral of sinh u du?
cosh u + C
What is the integral of sech^2 u du?
tanh u + C
What is the integral of csch^2 u du?
-coth u + C
What is the integral of sech u tanh u du?
-sech u + C
What is the integral of csch u coth u du?
-csch u + C
How do you find the particular solution?
Solve for C, then plug C back into the original equation (non-differential)
What is the initial condition?
The value of the dependent variable or one of its derivatives provided at initial time t=0. (y = 2 when x = -3)
What is the exponential growth and decay model?
y = Ce^kt, where C is the initial value of y, k is the proportionality constant. Growth occurs when k > 0 and decay occurs when k < 0.
For integrals of the form x^n e^ax, x^n sin ax, or x^n cos ax which function becomes u and which become dv?
x^n becomes u, dv = e^ax dx, sin ax dx, or cos ax dx
For integrals of the form x^n ln x dx, x^n arcsin ax dx, or x^n arctan ax dx which function become u and which becomes dv?
u = ln x, arcsin ax, or arctan ax and dv = x^n dx
For integrals of the form e^ax sin bx or e^ax cos bx, which function becomes u and which becomes dv?
u = sin bx or cox bx and let dv = e^ax dx
If the power of the sine is odd and positive, what do you do?
Save one sine factor and convert the remaining factors to cosine. Then expand and integrate.
If the power of the cosine is odd and positive, what do you do?
Save one cosine factor and convert the remaining factors to sines. Then expand and integrate.
If the powers of both the sine and cosine are even and nonnegative, what do you do?
Make repeated use of the power reducing identities
When using tabular integration and you return to the original function in the derivative column, what do?
Instead of going one column down on the right, go straight across and add an integral sign. Integrate this term.
When integrating trig integrals, do you want the degree of sin to be even or odd? Cos even or odd? If both the same, even or odd?
Sin - odd, cos - odd, both - odd
What are the steps to integrate sin or cos integrals?
- Factor out either sin x or cos x, whichever has the odd power. 2. Using pythagorean theorem, change the remaining factors to the other cofunction. 3. Let u = remaining cofunction (u = sin x or u = cos x) 4. Integrate
If the power of the secant is even and positive, what do?
Save a secant squared factor and convert the remaining factors to tangent. Then expand and integrate.
If the power of the tangent is odd and positive, what do?
Save a secant-tangent factor and convert the remaining factors to secants. Then expand and integrate.
If there are no secant factors and the power of tangent is even and positive, what do?
Convert a tangent-squared factor to a secant-squared factor, then expand and repeat if necessary
If the integral is of the form sec^m dx, where m is odd and positive, what do?
Use integration by parts, using sec^2 x as dv and sec x as u.
If working with trig integrals with powers of secant and tangent and none of the rules work, what do?
Convert to sine and cosine.
What is sin mx sin nx?
1/2 (cos (m-n)x - cos (m+n)x)
What is sin mx cos nx?
1/2 (sin (m-n)x + sin (m+n)x)
What is cos mx cos nx?
1/2 (cos (m-n)x + cos (m+n)x)
What is the integral of sqrt (a^2 - u^2) du?
1/2 (a^2 arcsin u/a + u*sqrt(a^2 -u^2)) + C
What is the integral of sqrt (u^2 - a^2) du?
1/2 (usqrt(u^2 - a^2) - a^2ln |u + sqrt(u^2 - a^2)|) + C where u > a.
What is the integral of sqrt(u^2 + a^2) du?
1/2 (u*sqrt(u^2 + a^2) + a^2 * ln |u + sqrt(u^2 + a^2)|) + C
What is the sin^2 u power reducing formula?
1 - cos 2u / 2
What is the cos^2 u power reducing formula?
1 + cos 2u / 2
What is the tan^2 u power reducing formula?
1 - cos 2u / 1 + cos 2u
