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