Question

Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.∫ dx / (x √(7 - x²))

Accepted Answer

Recognize that the integral has the form $$\int \frac{dx}{x \sqrt{a - x^2$}}$$, where $$a = 7. $$This suggests using a standard integral formula involving expressions of the form $$\frac{1}{x \sqrt{a - x^2$}}. $$Recall from the table of integrals that $$\int \frac{dx}{x \sqrt{a - x^2$}}$$ can be evaluated using a substitution or a known formula, often involving inverse trigonometric or logarithmic functions. Consider the substitution $$x = \sqrt{7} \sin 	heta$$, which simplifies the square root term: $$\sqrt{7 - x^2$} = \sqrt{7 - 7 \sin^2$ 	heta} = \sqrt{7} \cos 	heta. $$This substitution will help rewrite the integral in terms of $$	heta. $$Rewrite $$dx in $$terms of $$d	heta$$: since $$x = \sqrt{7} \sin 	heta$$, then $$dx = \sqrt{7} \cos 	heta \, d	heta. $$Substitute $$x$$, $$dx$$, and $$\sqrt{7 - x^2$}$$ into the integral to express it entirely in terms of $$	heta. $$Simplify the integral after substitution and then integrate with respect to $$	heta. $$Finally, revert back to the variable $$x$$ using the inverse substitution $$	heta = \arcsin \left( \frac{x}{\sqrt{7}} ight) to $$express the answer in terms of $$x.$$

Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ dx / (x √(7 - x²))

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Concetti chiave

Integration Using Standard Integral Forms

Trigonometric Substitution

Integrals Involving Rational Functions and Radicals