TBIL-Cal Integration of Trigonometry (TI3)

If $\sin$ ’s power is odd, rewrite the integral as $\int g (\cos (x)) \sin (x) d x$ and use $u = \cos (x) .$
If $\cos$ ’s power is odd, rewrite the integral as $\int h (\sin (x)) \cos (x) d x$ and use $u = \sin (x) .$

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Activity 5.3.8.

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Let’s consider

\int \sin^{2} (x) d x .

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(a)

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Use the fact that

\sin^{2} (θ) = \frac{1 - \cos (2 θ)}{2}

to rewrite the integrand using the above identities as an integral involving

\cos (2 x) .

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(b)

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Show that the integral evaluates to

\frac{1}{2} x - \frac{1}{4} \sin (2 x) + C .

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Activity 5.3.9.

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Let’s consider

\int \sin^{2} (x) \cos^{2} (x) d x .

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(a)

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Use the fact that

\cos^{2} (θ) = \frac{1 + \cos (2 θ)}{2}

and

\sin^{2} (θ) = \frac{1 - \cos (2 θ)}{2}

to rewrite the integrand using the above identities as an integral involving

\cos^{2} (2 x) .

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(b)

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Use the above identities to rewrite this new integrand as one involving

\cos (4 x) .

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(c)

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Show that integral evaluates to

\frac{1}{8} x - \frac{1}{32} \sin (4 x) + C .

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Activity 5.3.10.

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Consider

\int \sin^{4} (x) \cos^{4} (x) d x .

Which would be the most useful way to rewrite the integral?

$\int (1 - \cos^{2} (x))^{2} \cos^{4} (x) d x$
$\int \sin^{4} (x) (1 - \sin^{2} (x))^{2} d x$
$\int {(\frac{1 - \cos (2 x)}{2})}^{2} {(\frac{1 + \cos (2 x)}{2})}^{2} d x$

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Activity 5.3.11.

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Consider

\int \sin^{3} (x) \cos^{5} (x) d x .

Which would be the most useful way to rewrite the integral?

$\int (1 - \cos^{2} (x)) \cos^{5} (x) \sin (x) d x$
$\int \sin^{3} (x) {(\frac{1 + \cos (2 x)}{2})}^{2} \cos (x) d x$
$\int \sin^{3} (x) (1 - \sin^{2} (x))^{2} \cos (x) d x$

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Remark 5.3.12.

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We might also use some other trigonometric identities to manipulate our integrands, listed in Appendix B.

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Activity 5.3.13.

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Consider

\int \sin (θ) \sin (3 θ) d θ .

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(a)

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Find an identity from Appendix B which could be used to transform our integrand.

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(b)

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Rewrite the integrand using the selected identity.

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(c)

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Evaluate the integral.

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Subsection 5.3.2 Videos

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Figure 108. Video: Compute integrals involving products of trigonometric functions

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Subsection 5.3.3 Exercises

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Exercises available at https://tbil.org/calculus/preview/exercises/#/bank/TI3/

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