Integral calculation. Indefinite integral

Differentiation Rules

If f(x) = x6 +4x2 – 18x + 90

f’(x) = 6x5 + 8x – 18

*multiply by the power, than subtract one from the power.

Chain rule in transcendental

Take for example

y=esin(x)….let sin(x) = u gives; du/dx = cos(x)

y=eu….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.

Integration

Anti-differentiation is known as integration

The general indefinite formula is shown below,

Integration

FORMULAS FOR INTEGRATIONGENERAL Formulae

Trigonometric Formulae

Exponential and Logarithmic Formulae

Linear bracket Formula

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

Definite integrals

1 3 x

yy = x2 – 2x + 5 Area under curve = A

A = ∫1 (x2-2x+5) dx = [x3/3 – x2 + 5x]1

= (15) – (4 1/3) = 10 2/3 units2

3

3

Area under curves – signed area

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

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

y = 25 - x2

5

5

5

Area Between 2 curves continued…

If 2 curves pass through eachother multiple times than you must split up the integrands.

y1y2

C D

A = ∫C(y1-y2)dx + ∫0(y1-y2)dx D0A1 A2

Let A be total bounded by the curves y1 and y2 area,

thus;

A = A1 +A2

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.

y

y = x2 – 2x + 5

x

y = x2 – 2x + 5

x

LEFT

RIGHT