Question

Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.∫ cos(θ / 2) cos(7θ) dθ

Accepted Answer

Recognize that the integral involves the product of two cosine functions: $$\cos\left(\frac{	heta}{2}ight)$$ and $$\cos(7	heta). To $$simplify this, recall the product-to-sum identity for cosine: $$\cos A \cos B = \frac{1}{2} [\cos(A - B) + \cos(A + B)]. $$Apply the product-to-sum formula by letting $$A = \frac{	heta}{2}$$ and $$B = 7	heta. $$Substitute these into the identity to rewrite the integrand as $$\frac{1}{2} \left[ \cos\left(\frac{	heta}{2} - 7	hetaight) + \cos\left(\frac{	heta}{2} + 7	hetaight) ight]. $$Simplify the arguments inside the cosine functions: $$\frac{	heta}{2} - 7	heta = \frac{	heta}{2} - \frac{14	heta}{2} = -\frac{13	heta}{2}$$ and $$\frac{	heta}{2} + 7	heta = \frac{	heta}{2} + \frac{14	heta}{2} = \frac{15	heta}{2}. $$Rewrite the integral as $$\int \cos\left(\frac{	heta}{2}ight) \cos(7	heta) \, d	heta = \frac{1}{2} \int \left[ \cos\left(-\frac{13	heta}{2}ight) + \cos\left(\frac{15	heta}{2}ight) ight] d	heta. $$Remember that $$\cos(-x) = \cos x$$, so you can simplify the first cosine term. Split the integral into two separate integrals: $$\frac{1}{2} \left( \int \cos\left(\frac{13	heta}{2}ight) d	heta + \int \cos\left(\frac{15	heta}{2}ight) d	heta ight). $$Use the table of integrals to find the antiderivatives of each cosine term, which generally have the form $$\int \cos(k	heta) d	heta = \frac{1}{k} \sin(k	heta) + C.$$

Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ cos(θ / 2) cos(7θ) dθ

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Concetti chiave

Trigonometric Identities

Use of Integral Tables

Integration of Trigonometric Functions