7.2 Trigonometric Integrals

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TECHNIQUES OF INTEGRATION
7.2Trigonometric Integrals
In this section, we will learn How to use
trigonometric identities to integrate certain
combinations of trigonometric functions.
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TRIGONOMETRIC INTEGRALS

• We start with powers of sine and cosine.

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SINE COSINE INTEGRALS
Example 1

• Evaluate ? cos3x dx
• Simply substituting u cos x isnt helpful,
  since then du -sin x dx.
• In order to integrate powers of cosine, we would
  need an extra sin x factor.
• Similarly, a power of sine would require an
  extra cos x factor.

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SINE COSINE INTEGRALS
Example 1

• Thus, here we can separate one cosine factor and
  convert the remaining cos2x factor to an
  expression involving sine using the identity
  sin2x cos2x 1
• cos3x cos2x . cosx (1 - sin2x) cosx

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SINE COSINE INTEGRALS
Example 1

• We can then evaluate the integral by substituting
  u sin x.
• So, du cos x dx and

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SINE COSINE INTEGRALS

• In general, we try to write an integrand
  involving powers of sine and cosine in a form
  where we have only one sine factor.
• The remainder of the expression can be in terms
  of cosine.

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SINE COSINE INTEGRALS

• We could also try only one cosine factor.
• The remainder of the expression can be in terms
  of sine.

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SINE COSINE INTEGRALS

• The identity sin2x cos2x 1 enables us
  to convert back and forth between even powers of
  sine and cosine.

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SINE COSINE INTEGRALS
Example 2

• Find ? sin5x cos2x dx
• We could convert cos2x to 1 sin2x.
• However, we would be left with an expression in
  terms of sin x with no extra cos x factor.

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SINE COSINE INTEGRALS
Example 2

• Instead, we separate a single sine factor and
  rewrite the remaining sin4x factor in terms of
  cos x.
• So, we have

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SINE COSINE INTEGRALS
Example 2

• Substituting u cos x, we have du sin x dx.
• So,

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SINE COSINE INTEGRALS

• The figure shows the graphs of the integrand
  sin5x cos2x in Example 2 and its indefinite
  integral (with C 0).

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SINE COSINE INTEGRALS

• In the preceding examples, an odd power of sine
  or cosine enabled us to separate a single factor
  and convert the remaining even power.
• If the integrand contains even powers of both
  sine and cosine, this strategy fails.

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SINE COSINE INTEGRALS

• In that case, we can take advantage of the
  following half-angle identities

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SINE COSINE INTEGRALS
Example 3

• Evaluate
• If we write sin2x 1 - cos2x, the integral is
  no simpler to evaluate.

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SINE COSINE INTEGRALS
Example 3

• However, using the half-angle formula for sin2x,
  we have

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SINE COSINE INTEGRALS
Example 3

• Notice that we mentally made the substitution u
  2x when integrating cos 2x.
• Another method for evaluating this integral is
  using the reduction formula.

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SINE COSINE INTEGRALS
Example 4

• Find ? sin4x dx
• We could evaluate this integral using the
  reduction formula for ? sinnx dx.

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SINE COSINE INTEGRALS
Example 4

• However, a better method is to write and use a
  half-angle formula

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SINE COSINE INTEGRALS
Example 4

• As cos2 2x occurs, we must use another half-angle
  formula

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SINE COSINE INTEGRALS
Example 4

• This gives

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SINE COSINE INTEGRALS

• To summarize, we list guidelines to follow when
  evaluating integrals of the form ? sinmx
  cosnx dx
• where m 0 and n 0 are integers.

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STRATEGY A

• If the power of cosine is odd (n 2k 1), save
  one cosine factor.
• Use cos2x 1 - sin2x to express the remaining
  factors in terms of sine
• Then, substitute u sin x.

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STRATEGY B

• If the power of sine is odd (m 2k 1), save
  one sine factor.
• Use sin2x 1 - cos2x to express the remaining
  factors in terms of cosine
• Then, substitute u cos x.

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STRATEGIES

• Note that, if the powers of both sine and cosine
  are odd, either (A) or (B) can be used.

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STRATEGY C

• If the powers of both sine and cosine are even,
  use the half-angle identities
• Sometimes, it is helpful to use the identity

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TANGENT SECANT INTEGRALS

• We can use a similar strategy to evaluate
  integrals of the form ? tanmx secnx dx

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TANGENT SECANT INTEGRALS

• As (d/dx)tan x sec2x, we can separate a sec2x
  factor.
• Then, we convert the remaining (even) power of
  secant to an expression involving tangent using
  the identity sec2x 1 tan2x.

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TANGENT SECANT INTEGRALS

• Alternately, as (d/dx) sec x sec x tan x, we
  can separate a sec x tan x factor and convert
  the remaining (even) power of tangent to secant.

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TANGENT SECANT INTEGRALS
Example 5

• Evaluate ? tan6x sec4x dx
• If we separate one sec2x factor, we can express
  the remaining sec2x factor in terms of tangent
  using the identity sec2x 1 tan2x.
• Then, we can evaluate the integral by
  substituting u tan x so that du sec2x dx.

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TANGENT SECANT INTEGRALS
Example 5

• We have

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TANGENT SECANT INTEGRALS
Example 6

• Find ? tan5 ? sec7?
• If we separate a sec2? factor, as in the
  preceding example, we are left with a sec5?
  factor.
• This isnt easily converted to tangent.

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TANGENT SECANT INTEGRALS
Example 6

• However, if we separate a sec ? tan ? factor, we
  can convert the remaining power of tangent to an
  expression involving only secant.
• We can use the identity tan2? sec2? 1.

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TANGENT SECANT INTEGRALS
Example 6

• We can then evaluate the integral by substituting
  u sec ?, so du sec ? tan ? d?

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TANGENT SECANT INTEGRALS

• The preceding examples demonstrate strategies for
  evaluating integrals in the form ? tanmx secnx
  for two caseswhich we summarize here.

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STRATEGY A

• If the power of secant is even (n 2k, k 2)
  save sec2x.
• Then, use tan2x 1 sec2x to express the
  remaining factors in terms of tan x
• Then, substitute u tan x.

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STRATEGY B

• If the power of tangent is odd (m 2k 1), save
  sec x tan x.
• Then, use tan2x sec2x 1 to express the
  remaining factors in terms of sec x
• Then, substitute u sec x.

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OTHER INTEGRALS

• For other cases, the guidelines are not as
  clear-cut.
• We may need to use
• Identities
• Integration by parts
• A little ingenuity

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TANGENT SECANT INTEGRALS

• We will need to be able to integrate tan x by
  using Formula 5 from Section 5.5

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TANGENT SECANT INTEGRALS
Formula 1

• We will also need the indefinite integral of
  secant

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TANGENT SECANT INTEGRALS

• We could verify Formula 1 by differentiating the
  right side, or as follows.

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TANGENT SECANT INTEGRALS

• First, we multiply numerator and denominator by
  sec x tan x

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TANGENT SECANT INTEGRALS

• If we substitute u sec x tan x, then du
  (sec x tan x sec2x).
• The integral becomes ? (1/u) du ln u C

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TANGENT SECANT INTEGRALS

• Thus, we have

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TANGENT SECANT INTEGRALS
Example 7

• Find ? tan3x dx
• Here, only tan x occurs.
• So, we rewrite a tan2x factor in terms of sec2x.

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TANGENT SECANT INTEGRALS
Example 7

• Hence, we use tan2x - sec2x 1.
• In the first integral, we mentally substituted u
  tan x so that du sec2x dx.

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TANGENT SECANT INTEGRALS

• If an even power of tangent appears with an odd
  power of secant, it is helpful to express the
  integrand completely in terms of sec x.
• Powers of sec x may require integration by parts,
  as shown in the following example.

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TANGENT SECANT INTEGRALS
Example 8

• Find ? sec3x dx
• Here, we integrate by parts with

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TANGENT SECANT INTEGRALS
Example 8

• Then,

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TANGENT SECANT INTEGRALS
Example 8

• Using Formula 1 and solving for the required
  integral, we get

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TANGENT SECANT INTEGRALS

• Integrals such as the one in the example may seem
  very special.
• However, they occur frequently in applications of
  integration.

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COTANGENT COSECANT INTEGRALS

• Integrals of the form ? cotmx cscnx dx can be
  found by similar methods.
• We have to make use of the identity
  1 cot2x csc2x

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OTHER INTEGRALS

• Finally, we can make use of another set of
  trigonometric identities, as follows.

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OTHER INTEGRALS
Equation 2

• In order to evaluate the integral, use the
  corresponding identity.

Integral Identity
a ? sin mx cos nx dx
b ? sin mx sin nx dx
c ? cos mx cos nx dx
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TRIGONOMETRIC INTEGRALS
Example 9

• Evaluate ? sin 4x cos 5x dx
• This could be evaluated using integration by
  parts.
• Its easier to use the identity in Equation 2(a)

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