Integral calculation. Indefinite integral

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Integral calculation. Indefinite integral
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Differentiation Rules
If f(x) x6 4x2 18x 90 f(x) 6x5 8x
18 multiply by the power, than subtract one
from the power.
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Chain rule in transcendental
Take for example yesin(x).let sin(x) u gives
du/dx cos(x) yeu.dy/du eu Use chain rule
formula dy/dx eu.cos(x)
cos(x).esin(x) Thus, assign the function that is
inside of another function u, in this case
sin(x) in inside the exponential.
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Integration
Anti-differentiation is known as integration The
general indefinite formula is shown below,
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Integration
FORMULAS FOR INTEGRATION
GENERAL Formulae
Exponential and Logarithmic Formulae
Linear bracket Formula
Trigonometric Formulae
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Indefinite integrals Examples

• ? x5 3x2 dx x6/6 x3 c
• ? 2sin (x/3) dx 2 ? sin(x/3) dx -2x3cos(x/3)
  c
• ? x-2 dx -x-1 c
• ? e2x dx ½ e2x c
• ? 20 dx 20x c

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Definite integrals
y
y x2 2x 5
Area under curve A A ?1 (x2-2x5) dx
x3/3 x2 5x1 (15) (4 1/3) 10 2/3
units2
3
3
1 3 x
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Area under curves signed area
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Area Between 2 curves
Area Between two curves is found by subtracting
the Area of the upper curve by Area of the lower
curve. This can be simplified into Area ?
(upper curve lower curve) dx
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A ?-5 25-x2-(x2-25) dx OR A 2 ?0
25-x2-(x2-25) dx OR A 4 ?0 25-x2 dx A 83 1/3
units2
y x2 -25
5
5
y 25 - x2
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Area Between 2 curves continued
If 2 curves pass through eachother multiple times
than you must split up the integrands.
y2
y1
Let A be total bounded by the curves y1 and y2
area, thus A A1 A2
C
D
A1
A2
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Integration Area Approximation
The area under a curve can be estimated by
dividing the area into rectangles.

Two types of which is
the Left endpoint and right endpoint
approximations. The average of the left and
right end point methods gives the trapezoidal
estimate.
LEFT
y
RIGHT
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