Solving an Integral Using Tangent Substitution

Using tangent substitution, where $5x=2\tan(\theta)$, solve an integral with a coefficient on $x^2$.

Tangent substitution is one of the powerful techniques used in solving certain types of integrals, especially those involving roots and quadratic expressions. This technique leverages the identities from trigonometry to simplify integrals that can be quite challenging otherwise. In this particular problem, we're given a substitution equation where 5 times the variable x equals two times the tangent of theta. This helps to transform an algebraic expression involving x into a trigonometric expression involving theta, which can be more manageable to integrate.

A central concept here is recognizing the type of integral you're dealing with and identifying the appropriate trigonometric substitution that can simplify the integration process. For integrals with expressions like the square root of a quadratic, such substitution often simplifies the integrand to a more straightforward trigonometric function. By transforming the variable, the integration often reduces to a fundamental integral involving basic trigonometric functions, paving the way for easier computation.

After performing the integral in terms of theta, recalling the original substitution allows one to transform the result back into the variable x. This step is crucial to ensure the final solution is in the original variable's terms, providing a complete solution to the given integral problem. Recognizing the symmetry and implications of trigonometric identities is essential in effectively applying tangent substitution to integration.

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