The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ (dt / t√(3 + t²)

Verified step by step guidance

1. Identify the integral to solve: \(\int \frac{dt}{t \sqrt{3 + t^{2}}}\).
2. Recognize that the integrand contains \(t\) in the denominator and a square root of a quadratic expression \(3 + t^{2}\). This suggests using a substitution related to \(t^{2}\) or a trigonometric substitution to simplify the square root.
3. Consider the substitution \(t = \sqrt{3} \tan{\theta}\), which transforms \(3 + t^{2}\) into \(3 + 3 \tan^{2}{\theta} = 3 \sec^{2}{\theta}\). This substitution will simplify the square root expression.
4. Compute \(dt\) in terms of \(d\theta\): since \(t = \sqrt{3} \tan{\theta}\), then \(dt = \sqrt{3} \sec^{2}{\theta} d\theta\). Substitute \(t\) and \(dt\) back into the integral and simplify the expression.
5. Rewrite the integral entirely in terms of \(\theta\), simplify the integrand, and then integrate with respect to \(\theta\). After integration, substitute back \(\theta = \arctan{\left( \frac{t}{\sqrt{3}} \right)}\) to express the answer in terms of \(t\).

Key Concepts

Integration by Substitution

Integration by substitution involves changing variables to simplify an integral. By letting a new variable represent a function inside the integral, the integral can often be transformed into a more manageable form. This method is especially useful when the integral contains composite functions.

Trigonometric Substitution

Trigonometric substitution replaces algebraic expressions involving square roots with trigonometric functions to simplify integration. For integrals containing expressions like √(a² + x²), substituting x = a tan(θ) can transform the integral into a trigonometric integral that is easier to evaluate.

Algebraic Manipulation of Integrals

Algebraic manipulation involves rewriting the integrand using algebraic identities or simplifications before integrating. This can include factoring, rationalizing, or splitting the integral into simpler parts, making the integral more straightforward to solve.