Question

Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.∫ √(x - x²) / x dx

Accepted Answer

Start by rewriting the integrand to a more manageable form. Notice that the expression under the square root is $$x - x^2$, which can be factored as x(1 - x$$)$$. So the integral becomes $$\int \frac{\sqrt{x(1 - x)}}{x} \, dx$$. Simplify the integrand by separating the square root and dividing by x$: $$\frac{\sqrt{x(1 - x)}}{x} = \frac{\sqrt{x} \sqrt{1 - x}}{x} = \frac{\sqrt{1 - x}}{\sqrt{x}}$$. Rewrite the integral as $$\int \frac{\sqrt{1 - x}}{\sqrt{x}} \, dx = \int \frac{(1 - $$x)^{1/2}$$}{$$x^{1/2}$$} \, dx = \int $$x^{-1/2} (1$$ - $$x)^{1/2}$$ \, dx$$. To evaluate this integral, consider a substitution that simplifies the expression. A common substitution for integrals involving $$\sqrt{1 - x}$$ is x$$ = \$$sin^2$$ 	heta$$, because 1$$ - \$$sin^2$$ 	heta = \$$cos^2$$ 	heta$$. This substitution will transform the integral into a trigonometric integral. After substituting x$$ = \$$sin^2$$ 	heta$$, express dx$ in terms of d$$	heta$$, rewrite the integral entirely in terms of $$	heta$$, and then use the table of integrals to evaluate the resulting trigonometric integral.

Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ √(x - x²) / x dx

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Key Concepts

Integration Using Substitution

Using Integral Tables

Simplifying the Integrand