C4 Integration Techniques

Summary of the techniques used in integration

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Integration of e^x and 1/x

e^x

ne^kx => n/k.e^kx

1/x

n/kx => n/k.ln|x|

1/nx+k => 1/n . ln|nx+k|

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Integration of sin and cos

sinx => -cosx + c

cosx => sinx + c

sec^2x => tanx + c

-------------------------------------------------------------------

cosecxcotx => -cosecx + c

secxtanx => secx + c

cosec^2x => -cotx + c

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Integration of f'(x)/f(x)

If the differential is on top then it becomes ln of the denominator

~f'(x)/f(x) => ln|f(x)|

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Integration by Substitution

Allow something to equal 'u'

Integrate

Substitute the 'u' value back in at the end

-DO NOT FORGET to change dx to du by differentiating u and rearranging

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Integration by Parts

This is the reverse of the product rule

ʃ(u.dv/dx)dx => uv - ʃ(v.du/dx)dx

This is how you would integrate ln|x|

  • let u = lnx
  • let dv/dx = 1
  • find du/dx and v to use above formula
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Tough Integrals

REMEMBER YOU CAN USE:

-Trigonometric Identities

-Partial Fractions

-Some integrals are given in the formula booklet

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Volumes of Revolution

V = πʃ(y^2) dx with x limits

FOR PARAMETRIC

V = πʃ(y.dx/dt) dt with t limits

They may not give the limits as t so you will need to change them

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Differential Equations

Get x's and y's on the same side

dy/dx = yx => 1/y dy = x dx

DON'T FORGET after integrating both sides, put C on one of them (c is a constant so doesn't need to be on both sides)

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Trapezium Rule

h/2[y0+yn+2(y1+y2+...yn-1)]

h is width of strip, h=(b-a)/n

GIVEN FORMULA

% ERROR

=[(exact value-approximate value)/exact value] x 100

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