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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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.

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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.

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