Maths Integration

Maths Integration

1. Simplify the Integrand, if possible

   • 1:General Power Formula
     • Apply to trigonometric, logarithmic and exponential functions with powers. It involves a substitution of the part with the power
       • Sin^(1/3)x cosx. [Substitution is u=sinx]
       • sin^(-1)4x/sqrt(1-16x^(2)). [Substitution is u=sin^(-1)4x]
       • (3+ln2x)^(3)/x. [Substitution is u=3+ln2x]
     • 2: Basic Logarithmic Form
       • Any function with the derivative on top and function on bottom. It involves a substitution of u as the bottom of the fraction.
         • 1/x(1+2lnx). [Substitution is u=1+2lnx]
         • 2x^(3)/x^(4)+1. [Substitution is u=x^(4)+1]
       • 3: The Exponential Form
         • 4: Basic Trigonometric Forms
           • If the function contains more than 1x before the trig substitute out the factor as u
             • csc2xcot2x. [Substitution is 2x]
           • 5: Other Trigonometric Forms
             • tanx
               • tanx. [Rewriting is sinx/cosx and then use basic logarithmic form]
               • If there is a product of sine and cosine and if at least one of the powers is odd
                 • Cos^2x + sin^2x =1
                 • If there is a product of sine and cosine and powers are even
                   • 2cos^2x= 1 + cos2x or 2sin^2x= 1-cos2x
             • 6: Inverse Trignometric Forms
               • u=f(x)
                 • 1/sqrt(49-x^2). [Need to get a squared value out of 49-7]
               • 7: Integration by Parts
                 • Product of two functions
                 • 7.1- It is possible that a function needs to be double integrated by parts
                 • 7.2- If it is circular then move the same parts across to the left hand side and divide by any constants
                 • 8: Trigonometric Substitution
                   • Must be able to rewrite the equation to get squared values and substitute in as a
                     • 1/(x^2+9)^(3/2). [Substitution is atanx, then the equation needs to be simplified using trig identities if necessary before integrating]
   • Complete the Square

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