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Calc II exam 1

Terms

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(cos(x))^2
(1/2)(1+cos(2x)) OR 1-(sin(x))^2
(csc(x))^2
1+ (cot(x))^2
(sin(x))^2
(1/2)(1-cos(2x)) OR 1-(cos(x))^2
(tan(x))^2
(sec(x))^2 -1
arc length formula
L = ~sqrt(1+(f'(x))^2) dx
a^2 - x^2
u = sin^(-1)(x/a)
conditions for integrating ~(sin(x))^m(cos(x))^n dx
1. Either n or m (or both) is an odd, positive interger 2. Both m and n are non-negative even intergers
conditions for integrating ~(tan(x))^m(sec(x))^n dx
1. m is an odd, positive interger 2. n is an even, non-negative interger
cos(2x)
cos(x)^2-sin(x)^2
Dx cos^(-1)(x)
-1/(sqrt(1-x^2))
Dx cot^(-1)(x)
-1/(1+x^2)
Dx csc^(-1)(x)
-1/(|x|sqrt(x^2)-1)
Dx sec^(-1)(x)
1/(|x|sqrt(x^2)-1)
Dx sin^(-1)(x)
1/(sqrt(1-x^2))
Dx tan^(-1)(x)
1/(1+x^2)
Integration by parts
~udv = uv - ~vdu
Separate (x^2+2)/(x^3)(x-2)^2(x^2+1) into partial fractions
(a/(x))+(b/(x^2))+(c/(x^3))+(d/(x-2))+(e/(x-2)^2)+((fx+g)/(x^2+1))
separate x/((x+1)^2) into partial fractions
(a/(x+1)) + (b/(x+1)^2))
simplified integration by parts
~(polynomial)(e^x, sin(x), cos(x)) differentiate each separately, then combine crosswise, alternating between positive and negative starting with negative
sin(2x)
2sin(x)cos(x)
Surface area of f(x) around x-axis
SA = ~2(pi)f(x)sqrt(1+(f'(x))^2) dx (a, b on x-axis)
Surface area of f(x) around y-axis
SA = ~2(pi)(x)sqrt(1+(f'(x))^2) dx (a, b, on x-axis)
Surface area of g(y) around x-axis
SA = ~2(pi)(y)sqrt(1+(g'(y))^2) dy (a, b, on y-axis)
Surface area of g(y) rotated around y-axis
SA = ~2(pi)g(y)sqrt(1+(g'(y))^2) dy (a, b on y-axis)
volume generated by revolving f(x) and g(x) around x-axis
v = ~(pi)[f(x)^2-g(x)^2] dx (a, b on x-axis)
volume generated by revolving f(x) around y-axis
v = ~2(pi)x(f(x)) dx (a, b on x-axis)
volume generated by revolving g(y) around x-axis
v = ~2(pi)y(g(y)) dx (a, b on y-axis)
volume generated by rotating f(x)around x-axis
v = ~(pi)f(x)^2 dx (a, b on x-axis)
volume generated by rotating g(y) around y-axis (dish)
v = ~(pi)g(y)^2 dy (a, b on y-axis)
x^2 + a^2
u = tan^(-1)(x/a)
x^2 - a^2
u = sec^(-1)(x/a)
~cos(kx) dx
(1/k)sin(kx) + C
~cot(x) dx
ln|sin(x)| + C OR -ln|csc(x)| + C
~csc(x) dx
-ln|csc(x) + cot(x)| + C
~sec(x) dx
ln|sec(x) + tan(x)| + C
~sin(kx) dx
(-1/k)cos(kx) + C
~tan(x) dx
-ln|cos(x)| + C OR ln|sec(x)| + C

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