Use the substitution u = tan x to evaluate the integral
∫ dx / (1 + sin² x).

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Start with the given integral: \(\int \frac{dx}{1 + \sin^{2} x}\).

Use the substitution \(u = \tan x\). Recall that \(\frac{du}{dx} = \sec^{2} x\), so \(dx = \frac{du}{\sec^{2} x}\).

Express \(\sin^{2} x\) and \(\sec^{2} x\) in terms of \(u\). Since \(u = \tan x\), we have \(\sin x = \frac{u}{\sqrt{1+u^{2}}}\) and \(\sec^{2} x = 1 + \tan^{2} x = 1 + u^{2}\).

Rewrite the integral in terms of \(u\): replace \(dx\) with \(\frac{du}{1+u^{2}}\) and \(\sin^{2} x\) with \(\frac{u^{2}}{1+u^{2}}\). Simplify the denominator \(1 + \sin^{2} x\) accordingly.

Simplify the integral to a rational function in \(u\) and then integrate with respect to \(u\). After integration, substitute back \(u = \tan x\) to express the answer in terms of \(x\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Trigonometric Substitution

Trigonometric substitution involves replacing a trigonometric function with a new variable to simplify an integral. In this problem, substituting u = tan x transforms the integral into a rational function of u, making it easier to integrate.

Pythagorean Identities

Pythagorean identities relate sine, cosine, and tangent functions, such as 1 + tan²x = sec²x. These identities are essential for rewriting expressions involving sin²x in terms of tan x and sec x, facilitating substitution and simplification.

Integration of Rational Functions

After substitution, the integral often becomes a rational function in terms of u. Understanding how to integrate rational functions, including techniques like partial fractions or recognizing standard forms, is crucial to solving the integral.

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