Integration using Tangent Substitution

Integrate $\displaystyle \frac{dx}{(x^2 + 2^2)^{3/2}}$ using the tangent substitution where $x = 2 \tan(\theta)$.

Trigonometric substitution is a powerful technique often used to evaluate integrals involving expressions with square roots. In this problem, we specifically use tangent substitution to simplify and solve the integral. The substitution of a tangent function is particularly useful when the integral involves a sum or difference of squares or when the variable squared is accompanied by a constant, like the x squared plus four raised to the power of three halves in this integral. A key aspect of this method is transforming the given algebraic expression into a simpler trigonometric form that is more manageable.

In the transformation process, when we let x equal two times the tangent of theta, the integral's complexity reduces significantly as it changes into an expression involving secant, a trigonometric function with well-known integrals. The differential dx is also expressed in terms of theta, aiding in the simplification of the integral. This approach not only makes the integral easier to handle but also highlights the interconnectedness of algebraic and trigonometric functions in calculus.

By using trigonometric substitution, students can grasp how we can convert complex algebraic expressions into forms that are more mathematically tractable using the properties of trigonometry. This technique is especially beneficial in understanding how certain functions interact and how they can be manipulated to shed light on their integration, which is an essential skill in higher mathematics and various applications ranging from physics to engineering.

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