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

∫₀^{π/2} 4^{sin x} cos x dx

검증된 단계별 안내

Recognize that the integral is of the form \(\int_0^{\frac{\pi}{2}} 4^{\sin x} \cos x \, dx\), where the integrand involves an exponential function with a trigonometric exponent and a cosine factor.

Recall that \(4^{\sin x}\) can be rewritten using the exponential and natural logarithm as \(e^{\sin x \cdot \ln 4}\), which might help in substitution.

Use substitution by letting \(u = \sin x\). Then, compute the differential \(du = \cos x \, dx\). This substitution transforms the integral into an integral in terms of \(u\).

Change the limits of integration according to the substitution: when \(x = 0\), \(u = \sin 0 = 0\); when \(x = \frac{\pi}{2}\), \(u = \sin \frac{\pi}{2} = 1\).

Rewrite the integral in terms of \(u\) as \(\int_0^1 4^u \, du\), which is a standard integral of the form \(\int a^u \, du\) and can be integrated using the formula \(\int a^u \, du = \frac{a^u}{\ln a} + C\).

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Definite Integrals

A definite integral calculates the net area under a curve between two specific limits. It is represented as ∫_a^b f(x) dx, where a and b are the lower and upper bounds. Evaluating definite integrals often involves finding an antiderivative and then applying the Fundamental Theorem of Calculus.

Substitution Method

The substitution method simplifies integrals by changing variables to transform the integral into a more manageable form. Typically, you set u equal to a function inside the integral, then rewrite dx in terms of du. This technique is especially useful when the integral contains a composite function.

Exponential Functions with Variable Exponents

Exponential functions like a^{f(x)} involve a constant base raised to a variable exponent. When integrating such functions, it helps to rewrite the expression using natural logarithms: a^{f(x)} = e^{f(x) \, ln(a)}. This allows the use of chain rule and substitution for integration.

29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.∫₀^{π/2} 4^{sin x} cos x dx