Evaluate the integrals in Exercises 47–68.

∫₀ ^π/2 5(sin x)³/² cos x dx

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Recognize that the integral is of the form \(\int_0^{\frac{\pi}{2}} 5 (\sin x)^{\frac{3}{2}} \cos x \, dx\). Notice that the integrand involves \(\sin x\) raised to a power and multiplied by \(\cos x\), which suggests a substitution involving \(\sin x\).

Let \(u = \sin x\). Then, compute the differential \(du = \cos x \, dx\). This substitution will simplify the integral because \(\cos x \, dx\) can be replaced by \(du\).

Rewrite the integral in terms of \(u\): the limits of integration change accordingly. When \(x = 0\), \(u = \sin 0 = 0\). When \(x = \frac{\pi}{2}\), \(u = \sin \frac{\pi}{2} = 1\). The integral becomes \(\int_0^1 5 u^{\frac{3}{2}} \, du\).

Integrate the function \(5 u^{\frac{3}{2}}\) with respect to \(u\). Recall the power rule for integration: \(\int u^n \, du = \frac{u^{n+1}}{n+1} + C\), where \(n \neq -1\).

After integrating, apply the new limits of integration from \(0\) to \(1\) to evaluate the definite integral.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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, a part of the integrand is set as a new variable u, and dx is expressed in terms of du. This technique is especially useful when the integral contains a composite function.

Trigonometric Functions and Powers

Integrals involving powers of sine and cosine often require special techniques, such as substitution or using trigonometric identities. Understanding how to manipulate expressions like (sin x)^(3/2) and recognizing patterns helps in simplifying the integral for easier evaluation.

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