Anti-differentiation + Integration

How do you antidifferentiate

dy/dx = 20x⁴

∫20x⁴ dx

keep coefficient and x

add one to the power and divide by the new power

add +c

4x⁵ + c

Integrand

∫3Cos3x dx

Change Cos to corresponding

keep coefficient, keep angle

divide by the diff of the angle

add + c

Sin3x +c

∫2e^2x dx

Keep everything

divide by the diff of the power

add a +c

When is the antidifferentiated function a log function?

when there is a fraction, with no x’s on the top and no x’s in the diff of the bottom

∫10/2x+7 dx

keep the top

muliplied by ln of the bottom

add a +c

all multiplied by 1 over the diff of the bottom

1/5.10.ln(5x-9) +c

What is speed?

ds/dt

the first derivative

What is acceleration?

d²s/dt²

the second derivative

What is integration?

area under a curve stopping at the axis

∫y dx when bringing it to the x axis

∫x dy

How do you integrate above the x-axis?

∫y dx

sub in y

integrate/differentiate y

put in limits to specify the area to be found

sub in the values of the limits

subtract the lower limit value subbed in from the higher limit value

How do you integrate above and below the x axis?

∫y dx

sub in y

integrate y

put in limits to specify the area to be found, integrate the above and below separately

sub in the values of the limits

subtract the lower limit value subbed in from the higher limit value

multiply the area under the x-axis by -1

add the areas together

How do you use Simpson’s Trapezoidal rule?

y_n is the last height

y_n-1 is the second last height

h is the distance between heights, and is a constant

sub in values to get approximate area

What is the formula for % error?

error(difference)/true value x 100/1 = % error