Question

Evaluate the integrals in Exercises 1–22.∫₀^π 8cos⁴(2πx) dx

Accepted Answer

Recognize that the integral is $$ \int_0^{\pi} 8 \cos^4(2\pi x) \, dx . $$The constant 8 can be factored out of the integral, so rewrite it as $$ 8 \int_0^{\pi} \cos^4(2\pi x) \, dx . $$Use the power-reduction formula to express $$ \cos^4(	heta)  in $$terms of cosines of multiple angles. Recall that $$ \cos^4(	heta) = \left( \cos^2(	heta) ight)^2 $$ and $$ \cos^2(	heta) = \frac{1 + \cos(2	heta)}{2} . $$Substitute $$ \cos^2(2\pi x) = \frac{1 + \cos(4\pi x)}{2} $$ into the expression and then square it to get $$ \cos^4(2\pi x) = \left( \frac{1 + \cos(4\pi x)}{2} ight)^2 . $$Expand the squared term to get $$ \cos^4(2\pi x) = \frac{1}{4} \left( 1 + 2\cos(4\pi x) + \cos^2(4\pi x) ight) . $$Then apply the power-reduction formula again to $$ \cos^2(4\pi x)  to $$express it in terms of cosines of multiple angles. Rewrite the integral as a sum of integrals of cosines with different frequencies, integrate each term over $$ 0  to$$ $$ \pi $$, and then multiply the result by 8. Remember to use the fact that $$ \int_0^{\pi} \cos(kx) \, dx = 0 $$ for nonzero integer multiples $$ k  of$$ $$ \pi .$$

Evaluate the integrals in Exercises 1–22.
∫₀^π 8cos⁴(2πx) dx

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Key Concepts

Definite Integrals

Trigonometric Power Reduction

Integration of Cosine Functions