29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.

∫₀^{π} 2^{sin x} · cos x dx

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Recognize that the integral is of the form \(\int_0^{\pi} 2^{\sin x} \cdot \cos x \, dx\), where the integrand involves an exponential function with a trigonometric exponent multiplied by \(\cos x\).

Use substitution to simplify the integral. Let \(u = \sin x\). Then, compute the differential \(du = \cos x \, dx\).

Rewrite the integral in terms of \(u\): since \(du = \cos x \, dx\), the integral becomes \(\int_{u=\sin 0}^{u=\sin \pi} 2^u \, du\).

Evaluate the new limits of integration: \(\sin 0 = 0\) and \(\sin \pi = 0\), so the integral becomes \(\int_0^0 2^u \, du\).

Notice that the limits of integration are the same, which implies the value of the integral is zero without further calculation.

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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 limits, here from 0 to π. It involves evaluating the integral of a function over a specified interval, resulting in a numerical value that represents accumulation or total change.

Integration by Substitution

Integration by substitution simplifies integrals by changing variables to transform the integral into a more manageable form. It is especially useful when the integrand contains a function and its derivative, allowing easier evaluation.

Properties of Exponential Functions with Variable Exponents

Functions like 2^(sin x) involve an exponential with a variable exponent. Understanding how to handle such expressions, especially when combined with trigonometric functions, is key to setting up substitutions and simplifying the integral.

29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.∫₀^{π} 2^{sin x} · cos x dx