MATH 2414 Exam 1 Flashcards

d/dx [sinh(u)]

cosh(u)

d/dx [cosh(u)]

sinh(u)

d/dx [tanh(u)]

sech²(u)

d/dx [sech(u)]

-sech(u)tanh(u)

d/dx [csch(u)]

-csch(u)coth(u)

d/dx [coth(u)]

-csch²(u)

Integral of [sinh(u) du]

cosh(u) + C

Integral of [cosh(u) du]

sinh(u) + C

Integral of [sech²(u) du]

tanh(u) + C

Integral of [sech(u)tanh(u) du]

-sech(u) + C

Integral of [csch(u)coth(u) du]

-csch(u) + C

Integral of [csch²(u) du]

-coth(u) + C

sinh(x)

(e^x - e^-x)/2

cosh(x)

(e^x + e^-x)/2

Fundamental hyperbola identity

cosh²x - sinh²x = 1

Double Angle Identity for sine

sin²x = ½ - ½cos2x

Double Angle Identity for cosine

cos²x = ½ + ½cos2×

Integral of [cos(ax)dx]

1/a * sin(ax) + C

Integral of [sin(ax) dx]

1/a * cos(ax) + C

Integral of [sec²(ax) dx]

1/a * tan(ax) + C

Integral of [sec(ax)tan(ax) dx]

1/a * sec(ax) + C

Integral of [csc(ax)cot(ax) dx]

-1/a * csc(ax) + C

Integral of [csc²(ax) dx]

-1/a * cot(ax) + C

Integral of [dx/sqrt(a²+x²)]

sin^-1(x/a) + C

Integral of [dx/(a²-x²)]

1/a tan^01x/a) + C

Integral of [dx/(x * sqrt(x²+a²))]

1/a * sec^-1|x/a| + C

tanh(x)

sinh(x)/cosh(x)

sech(x)

1/cosh(x)

csch(x)

1/sinh(x)

coth(x)

1/tanh(x) = cosh(x)/sinh(x)

Integral of [tanh(x) dx]

ln(cosh(x)) + C

Integral of [coth(x) dx]

ln|sinh(x)| + C

Integral of [sech(x) dx]

tan^-1(sinh(x)) + C

Integral of [csch(x) dx]

ln|tanh(x/2)| + C

Area of a Region between Two Curves

A = Integral_a^b[f(x) - g(x) dx]

Area of a Region between Two Curves with respect to y

A = Integral_c^d[f(y) - g(y) dy]

Washer Method

V = Integral_a^b[pi{[f(x)]²-[g(x)]²} dx] — revolves around x-axis

V = Integral_c^d[pi{[f(y)]²-[g(y)]²} dy] — revolves around y-axis

Washer Method with a K

V = Integral_a^b[pi{[f(x) - k]²-[g(x) - k]²} dx] — revolves around x-axis

V = Integral_c^d[pi{[f(y) - k]²-[g(y) - k]²} dy] — revolves around y-axis

Shell Method

V = Integral_a^b2(pi)x[f(x) - g(x)] dx] — revolves around y-axis

V = Integral_c^d[2(pi)y[f(y) - g(y)] dy] — revolves around x-axis

Arc Length for y = f(x)

L = Integral_a^b[sqrt(1 + [f’(x)]²) dx]

L = Integral_c^d[sqrt(1 + [g’(y)]²) dy]

Formula for Work

W = Integral_a^b[F(x) dx]

Formula for Spring Problem

W = Integral_a^b[k(x) dx]

Formula for Lifting Problems

W = Integral_a^b[pg(b-y) dx]

Fluid Lifting Problems

W = Integral_a^b[pg * A(y)D(y) dy]

Force and Pressure

W = Integral₀^a[pg(b-y)w(y) dy]