Integral Evaluation Using Trigonometric Substitution

Evaluate the integral $\displaystyle \int \frac{dt}{t^2 + 9}$ using trigonometric substitution.

Trigonometric substitution is a powerful technique for evaluating integrals, particularly suited for integrals involving the sum or difference of squares under a square root. In this problem, the integral involves a sum of squares in the denominator, which suggests that a trigonometric substitution may simplify the expression. The trigonometric identities allow us to convert algebraic expressions into trigonometric forms, making the integration process more manageable. In particular, for integrals of the form involving $t^2 + a^2$, substituting $t$ with a tangent function of a new variable can simplify the integral dramatically.

The substitution leverages the identity $1 + \tan^2(\theta) = \sec^2(\theta)$, turning the denominator into something simpler that can often be integrated using basic trigonometric integrals. Once the substitution is made, it's often necessary to adjust the differential $dt$ using the derivative of the chosen trigonometric function. The resulting integral is usually a much simpler form that can be evaluated directly. After integration, back-substituting the original variable is crucial to express the solution in terms of the initial variable.

Understanding and mastering trigonometric substitution can significantly enhance your problem-solving toolkit for solving integrals. It forms a bridge between algebraic manipulation and geometric intuition, allowing for elegant solutions to otherwise complex integral problems. This method also helps in solidifying your understanding of trigonometric identities and their applications in calculus.

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