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Integration TechniquesThis integration technique is particularly useful for integrands involving products of algebraic and transcendental functions. It is based upon product rule formula. The choice of u and v are critical to the process.

• Trylettingdvbethemostcomplicatedportionoftheintegrandthatfitsa basic integration rule. Then u will be the remaining factor.• Try letting u be the portion of the integrand whose derivative is a simpler function. Then dv will be the remaining factor.

Summary of some common integrals using integration by parts 1. For integrals of the form:

Let u = xn and let dv = eax dx,sin, ax dx,or cos, ax dx

2. For integrals of the form:

Let u = ln x,sin-1 ax,or tan-1 ax and let dv = xn dx

3. For integrals of the form:

Let u = sin bx or cos bx and let dv = eax dx

Note: • It may be necessary to do integration by parts multiple times before arriving at an integral fittingabasicintegrationrule.Ifthisisnecessarybesuretoenclosesubsequentapplications inparenthesessoapplicablesignsandcoefficientscanbecorrectlydistributed.• When making repeated applications of integration by parts be careful not to interchange the substitutions. This will just cancel out what was previously done.• Watch for a constant multiple of the original integrand. Collect the like integrands and divide bytheresultingcoefficient.Thissituationariseswithintegralsofform3aboveandor functions such as where m is odd.

CAS w Math

Integration Techniques - Trig IntegralsThistechniqueisusedtofindintegralsoftheform orwhere either m or n is a positive integer. The idea is to break them into integrals to which the power rule can be applied.

Useful Trig Identities

Guidelines for Evaluating Integrals Involving Sine and Cosine 1. If the power of the sine is odd and positive, save one factor and convert the remaining factors to cosines. Expand and integrate.

2. If the power of the cosine is odd and positive, save one factor and convert the remaining factors to sines. Expand and integrate.

3. If powers of both sine and cosine are positive and even make repeated use of the identities involving cos 2x to convert the integral to odd powers of the cosine and proceed as in 2.

Guidelines for Evaluating Integrals Involving Secant and Tangent 1. If the power of the secant is even and positive, save a secant-squared factor and convert the remaining factors to tangents. Then expand and integrate.

2. If the power of the tangent is odd and positive, save a secant-tangent factor and convert the remain factors to secants. Then simplify and integrate.

3. If there are no secant factors and the power of the tangent is even and positive, convert a tangent-squared factor into a secant-squared factor, then expand and repeat as necessary.

4. If there are no tangent factors and the power of the secant is odd, use integration by parts 5. If none of these apply, try converting to sines and cosines.

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Integration Techniques - Trig SubstitutionTrig substitution is used to evaluate integrals involving radicals of the form (a2 - x2 ), (a2+ x2 ) and (x2 - a2). The objective of this method is to eliminate the radical by use of the Pythagorean identities.

Trigonometric Substitutions (a > 0)

• Make substitutions using the above table for reference. • Solve the resulting trig integral • Refer to the triangle associated with the substitution used to convert trig functions back in terms of x and a

CAS w Math

Integration Techniques - Partial FractionsThis technique is used to break rational functions into simpler rational functions to which basic integration formulas can be applied. It is practical only for integrals of rational fractions whose denominators factor “nicely”.

Once you have written the decomposition, multiply the equation through by the least common denominator (which should be the denominator of the original fraction). Because this equation will be true for all values of x, you can substitute in any convenient value of x to obtain values for A, B, C, etc. The most convenient values for x will be those that make a particular factor 0.

Example: Since this is a proper fraction we begin by factoring the denominator:

The decomposition would then include one fraction for each power of x and x+1:

Multiplying by the least common denominator we get: We will need to pick 3 values for x; x = 0 and x = - 1 since these make a factor 0 and x = 1 because it is easy to calculate.For x = 0 : 5(0) + 20(0) + 6 = A(1) + B(0) + C(0) 6 = AFor x = - 1 : 6(0) + B(0) + C(- 1) = 5(-1)2 +20(- 1) + 6 9 = CFor x = 1 : 6(2)2 + B(1)(2) +9(1) = 5(1)2 + 20(1) + 6 B = - 1

So =

Lawson, Hoffstetler, Edwards: Calculus Early Transcendental Functions