Question

Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.∫ dx / (x² √(x² + 1))

Accepted Answer

Identify the integral to solve: $$\int \frac{dx}{x$$^{2}$$ \sqrt{x$$^{2}$$ + 1}}. $$Recognize that the integrand contains $$\sqrt{x$$^{2}$$ + 1}$$, which suggests using a trigonometric substitution where $$x = 	an(	heta)$$ because $$1 + 	an$$^{2}$$(	heta) = \sec$$^{2}$$(	heta). $$Substitute $$x = 	an(	heta)$$, then compute $$dx = \sec$$^{2}$$(	heta) d	heta. $$Also, rewrite the integral in terms of $$	heta$$: replace $$x^{2}$$ with $$	an$$^{2}$$(	heta)$$ and $$\sqrt{x$$^{2}$$ + 1}$$ with $$\sqrt{	an$$^{2}$$(	heta) + 1} = \sec(	heta). $$Rewrite the integral as $$\int \frac{\sec$$^{2}$$(	heta) d	heta}{	an$$^{2}$$(	heta) \cdot \sec(	heta)} = \int \frac{\sec$$^{2}$$(	heta)}{	an$$^{2}$$(	heta) \sec(	heta)} d	heta = \int \frac{\sec(	heta)}{	an$$^{2}$$(	heta)} d	heta. $$Simplify the integrand and use trigonometric identities to express everything in terms of sine and cosine, then integrate with respect to $$	heta. $$After integration, substitute back $$	heta = \arctan(x) to $$express the answer in terms of $$x.$$

Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ dx / (x² √(x² + 1))

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Key Concepts

Trigonometric Substitution

Integration of Rational Functions

Fundamental Integration Techniques