Chapter 6: Techniques of Integration
Section 6.2: Trigonometric Integrals
Derive the first reduction formula in Table 6.2.1.
The derivation in Table 6.2.7(a) begins with an integration by parts in which
This creates a factor of sinn+2x in the new integral on the right, a factor that is replaced by 1−cos2x. This replacement then creates an integral on the right that is a multiple of the original integral on the left. Solving for the unknown integral results in the desired reduction formula.
=cosm−1xsinn+1xn+1−−m−1n+1 ∫cosm−2xsinn+2x dx
=cosm−1xsinn+1xn+1+m−1n+1 ∫cosm−2xsinnx1−cos2x dx
=cosm−1xsinn+1xn+1+m−1n+1 ∫cosm−2xsinnx dx−∫cosmxsinnx dx
=cosm−1xsinn+1xn+1+m−1n+1 ∫cosm−2xsinnx dx−m−1n+1 ∫cosmxsinnx dx
=cosm−1xsinn+1xn+1+m−1n+1 ∫cosm−2xsinnx dx
Table 6.2.7(a) Derivation of the first reduction formula in Table 6.2.1
Install the IntegrationTools package.
Assign the name Q to the given integral.
Integrate by parts
Massage to the form in line 1 of Table 6.2.7(a)
Replace sin2x with 1−cos2x and massage to form in line 4 of Table 6.2.7(a)
Solve for the unknown integral
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