Evaluate the integrals in Exercises 53–58.
∫ from -π/2 to π/2 of cos(x) cos(7x) dx

검증된 단계별 안내

Recognize that the integral involves the product of two cosine functions: \(\cos(x)\) and \(\cos(7x)\), integrated over the interval \([-\frac{\pi}{2}, \frac{\pi}{2}]\).

Use the product-to-sum identity for cosine functions: \(\cos A \cos B = \frac{1}{2} [\cos(A - B) + \cos(A + B)]\). Applying this, rewrite the integrand as \(\frac{1}{2} [\cos(x - 7x) + \cos(x + 7x)] = \frac{1}{2} [\cos(-6x) + \cos(8x)]\).

Since cosine is an even function, \(\cos(-6x) = \cos(6x)\), so the integrand simplifies to \(\frac{1}{2} [\cos(6x) + \cos(8x)]\).

Split the integral into two separate integrals: \(\frac{1}{2} \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos(6x) \, dx + \frac{1}{2} \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos(8x) \, dx\).

Evaluate each integral by finding the antiderivative of cosine, which is sine divided by the coefficient of \(x\), and then apply the limits of integration \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) for each term.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Definite Integrals

A definite integral calculates the net area under a curve between two specified limits. It is evaluated by finding the antiderivative of the integrand and then applying the Fundamental Theorem of Calculus to compute the difference at the upper and lower bounds.

Product-to-Sum Trigonometric Identities

These identities transform products of trigonometric functions into sums or differences, simplifying integration. For example, cos(A)cos(B) = ½[cos(A−B) + cos(A+B)], which helps break down complex integrals into easier terms.

Symmetry of Trigonometric Functions

Understanding whether a function is even, odd, or neither helps simplify definite integrals over symmetric intervals. Cosine is an even function, so integrals from -a to a can be simplified by doubling the integral from 0 to a.

Evaluate the integrals in Exercises 53–58.∫ from -π/2 to π/2 of cos(x) cos(7x) dx