Question

Evaluate the integrals. $$\int \frac{d x}{\sqrt{9+x^{2}}}$$

   Evaluate the integrals.
$$\int \frac{d x}{\sqrt{9+x^{2}}}$$

Calculus

George B. Thomas,… 12th Edition

Chapter 8, Problem 1

Instant Answer

Solved by Expert Abdul Vahid M

07/22/2020

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Let $x = 3\tan\theta$. Then, $dx = 3\sec^2\theta d\theta$. Show more…

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Evaluate the integrals. $$\int \frac{d x}{\sqrt{9+x^{2}}}$$

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Key Concepts

Trigonometric Substitution

Trigonometric substitution is a method used to simplify integrals that involve square roots of quadratic expressions. By substituting x with a trigonometric function (for example, x = a tan(?)) when the integrand contains sqrt(a² + x²), the square root is transformed into a trigonometric function, making the integral easier to solve.

Inverse Hyperbolic Functions

Inverse hyperbolic functions, like the inverse hyperbolic sine (arcsinh), are useful in expressing the antiderivative of certain integrals. For instance, the integral ?dx/?(a² + x²) can be represented as arcsinh(x/a), which is equivalent to using logarithmic expressions to express the antiderivative.

Standard Integration Formulas

Standard integration formulas provide a quick reference for the result of common integrals. One such formula is ?dx/?(a² + x²) = ln|x + ?(x² + a²)| + C, which encapsulates the result of applying trigonometric substitution or recognizing the inverse hyperbolic function representation.

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