In an x-y Cartesian coordinate system, equation of the circle is:
(x-a)2 + (y-b)2 = r2
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sin 0° = 0
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cos 0° = 1
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tan 0° = 0
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sin 30° = 1/2
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cos 30° = (√3)/2
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tan 30° = 1/(√3)
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sin 45° = 1/(√2)
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cos 45° = 1/(√2)
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tan 45° = 1
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sin 60° = (√3)/2
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cos 60° = 1/2
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tan 60° = √3
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sin 90° = 1
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cos 90° = 0
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tan 90° = ∞
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sinq = 1/cosecq
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cosq = 1/secq
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tanq = 1/cotq
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sinq/cosq = tanq
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sin2q + cos2q = 1
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1 + tan2q = sec2q
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1 + cot2q = cosec2q
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sin( 90° - q ) = cosq
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cos( 90° - q ) = sinq
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tan( 90° - q ) = cotq
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sin( 90° + q ) = cosq
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cos( 90° + q ) = - sinq
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tan( 90° + q ) = - cotq
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sin( 180° - q ) = sinq
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cos( 180° - q ) = - cosq
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tan( 180° - q ) = - tanq
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sin( 180° + q ) = - sinq
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cos( 180° + q ) = - cosq
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tan( 180° + q ) = tanq
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Pythagorean Theorem:
The square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the legs
a2 = b2 + c2
a,b - two sides of the triangle connected by the right angle
c - hypotenuse of the triangle
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circumference of a circle
circumference of a circle = 2 . π . r
where,
π = PI = 22/7
r = radius of circle
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Area of Triangle
Area of Triangle = (1/2) . b . h
where,
h = height of triangle
b = the length of the base of triangle
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Area of rectangle
Area of rectangle = l . b
where,
l = length of rectangle
b = width of rectangle
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Area of circle
Area of circle = π . r 2
where,
π = PI = 22/7
r = radius of circle
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Area of trapezoid
Area of trapezoid = (1/2) . (height). (base one + base two)
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Area of Ellipse
Area of Ellipse = π . r1 . r2
where,
r1 = major radius
r2 = minor radius
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Area of Cylinder (surface area)
Area of Cylinder (surface area) =
2 . π . r . h
where,
r = radius of cylinder
h = length of cylinder
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Area of Cone (surface area)
Area of Cone (surface area) =
π . r . l
where,
r = radius of cone
l = length of side of the cone
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Volume of cylinder
Volume of cylinder = π . r 2 . h
where,
π = PI = 22/7
r = radius of cylinder
h = length of cylinder
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Volume of sphere
Volume of sphere = (4/3) . π . r 3
where,
π = PI = 22/7
r = radius of sphere
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Volume of Cone
Volume of Cone = (1/3) . π . r 2 . h
where,
π = PI = 22/7
r = radius of Cone
h = height of cone
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sin(-x) = -sin(x)
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cosec(-x) = -cosec(x)
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cos(-x) = cos(x)
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sec(-x) = sec(x)
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tan(-x) = -tan(x)
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cot(-x) = -cot(x)
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sin( a + b ) = sina cosb + cosa sinb
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sin( a - b ) = sina cosb - cosa sinb
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cos( a + b ) = cosa cosb - sina sinb
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cos( a - b ) = cosa cosb + sina sinb
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tan( a + b ) =
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(tana + tanb)
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(1 - tana tanb)
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tan( a - b ) =
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(tana - tanb)
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(1 + tana tanb)
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sin2a = 2 sina cosa
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cos2a = cos2a - sin2a
= 2cos2a - 1
= 1 - 2sin2a
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tan2a =
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(2 tana)
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(1 - tan2a)
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sin3a = 3sina - 4sin3a
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cos3a = 4cos3a - 3cosa
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For any triangle ABC with side lengths a,b,c
Law of Cosines
c2 = a2 + b2 - 2 a b cos C
b2 = c2 + a2 - 2 c a cos B
a2 = b2 + c2 - 2 b c cos A
Law of Sines
sinA/a = sinB/b = sinC/c
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cos θ cos β =
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cos(θ - β) + cos(θ + β)
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2
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sin θ sin β =
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cos(θ - β) - cos(θ + β)
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2
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sin θ cos β =
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sin(θ + β) + sin(θ - β)
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2
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sin3 θ =
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3sin θ - sin 3θ
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4
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cos3 θ =
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3cos θ + cos 3θ
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4
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sin3 θ . cos3 θ =
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3sin 2θ - sin 6θ
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32
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sin4 θ =
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3 - 4 cos 2θ + cos 4θ
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8
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cos4 θ =
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3 + 4 cos 2θ + cos 4θ
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8
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sin4 θ . cos4 θ =
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3 - 4 cos 4θ + cos 8θ
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128
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sin5 θ =
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10 sin θ - 5 sin 3θ + sin 5θ
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16
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cos5 θ =
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10 cos θ + 5 cos 3θ + cos 5θ
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16
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sin5 θ . cos5 θ =
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10 sin 2θ - 5 sin 6θ + sin 10θ
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512
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Quadratic Equation
For the equation:
a x 2 + b x + c = 0
Quadratic equation solving calculator
the value of x will be
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x =
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- b ± √ (b2 - 4 a c)
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2 a
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(a + b)2 = a2 + 2 a b + b2
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(a - b)2 = a2 - 2 a b + b2
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(a + b) . (a - b) = (a2 - b2)
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Arithmetic progression:
Arithmetic progression is a sequence of numbers such that the difference of any two successive members
of the sequence is a constant.
For example: Suppose a1, a2, a3, a4, ...... , an-1, an
are in sequence of arithmetic progression
Then the first term of an arithmetic series is a1 and assume that the common difference of
successive members is d, then the nth term of the sequence is:
an = a1 + (n - 1).d
The sum of all the components of an arithmetic series is:
Sn = a1 + a2 + a3 + ....... + an-1 + an
i.e. Sn = (n).(a1 + an)/2
Calculator for solving arithmetic series
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Geometric progression:
geometric progression OR geometric series is a sequence of numbers such that the quotient of any two
successive members of the sequence is a constant. The ratio of two successive number is called common ratio.
The constant ratio must not be equal to 0.
Example of geometric series :
ar1, ar2, ar3, . ....... ., arn-1, arn
The nth term of the geometric series can be defined as:
an = a r(n - 1)
r is called common ratio and n must be greater than 0:
Calculator for solving geometric series
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Logarithms:
logb 1 = 0
logb b = 1
logb(X . Y) = logbX + logbY
logb(X / Y) = logbX - logbY
logb(Xn) = n . logbX
logmn . lognm = 1
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d(u + v)
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=
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du
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+
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dv
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dx
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dx
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dx
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d(u . v)
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= u
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dv
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+ v
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du
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dx
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dx
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dx
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d
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(
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u
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) =
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v (du/dx) - u (dv/dx)
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dx
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v
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v2
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d
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( u n )
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= n u n-1
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du
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dx
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dx
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d
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( C u )
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= C u ln(C)
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du
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dx
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dx
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d( sin(u))
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= cos(u)
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du
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dx
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dx
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d( cos(u))
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= - sin(u)
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du
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dx
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dx
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d( tan(u))
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= sec2(u)
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du
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dx
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dx
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d( cosec(u))
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= - cosec(u) . cot(u)
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du
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dx
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dx
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d( sec(u))
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= sec(u) . tan(u)
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du
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dx
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dx
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d( cot(u))
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= - cosec2(u)
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du
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dx
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dx
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d(sin-1u)
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=
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1
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du
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dx
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dx
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d(cos-1u)
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=
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-1
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du
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dx
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dx
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d(tan-1u)
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=
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1
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du
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dx
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1 + u2
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dx
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d(cot-1u)
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=
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- 1
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du
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dx
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1 + u2
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dx
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d(sec-1u)
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=
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1
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du
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dx
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|u|
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dx
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d(cosec-1u)
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=
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-1
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du
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dx
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|u|
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dx
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