First: Final Answer
The final answer to the given integral ∫[1 to 4] (√(x^2 + 1)) / (x * √x) dx using the trigonometric substitution x = tan(θ) is ∫[π/4 to arctan(4)] dθ.
Second: Explanation
To solve the integral ∫(√(x^2 + 1)) / (x * √x) dx, we can use the trigonometric substitution x = tan(θ). Taking the derivative of both sides with respect to x, we get dx = sec^2(θ) dθ. Substituting x = tan(θ) and dx = sec^2(θ) dθ into the integral, we obtain ∫(√(tan^2(θ) + 1)) / (tan(θ) * √tan(θ)) sec^2(θ) dθ.
Simplifying the expression inside the integral using the trigonometric identity tan^2(θ) + 1 = sec^2(θ), we get ∫(sec(θ)) / (tan(θ) * √tan(θ)) dθ. Further simplifying by writing tan(θ) as sin(θ)/cos(θ) and using the identity sec(θ) = 1/cos(θ), we arrive at the integral ∫(1 / (sin(θ) * √sin(θ) * cos^2(θ))) dθ.
Now, recognizing that sin(θ) = x/√(x^2 + 1) and cos(θ) = 1/√(x^2 + 1), we can substitute these expressions back into the integral, yielding ∫(√(x^2 + 1) / (x * √x)) dx = ∫(1 / (x * √x)) dx. The limits of integration also transform accordingly, becoming [π/4 to arctan(4)]. Thus, the final answer is ∫[π/4 to arctan(4)] dθ.