2–9. Integrals Evaluate the following integrals.


∫₁⁴ (10^{√x} / √x) dx

Verified step by step guidance

1. Identify the integral to be solved: \(\int_1^4 \frac{10^{\sqrt{x}}}{\sqrt{x}} \, dx\).
2. Use the substitution method by letting \(t = \sqrt{x}\), which implies \(x = t^2\).
3. Calculate the differential \(dx\) in terms of \(dt\): since \(x = t^2\), then \(dx = 2t \, dt\).
4. Rewrite the integral in terms of \(t\): substitute \(\sqrt{x} = t\) and \(dx = 2t \, dt\) into the integral, and adjust the limits accordingly (when \(x=1\), \(t=1\); when \(x=4\), \(t=2\)).
5. Simplify the integral expression after substitution and set it up for integration with respect to \(t\).

Key Concepts

Definite Integrals

A definite integral calculates the net area under a curve between two specific points, called limits of integration. It results in a numerical value representing the accumulation of quantities, such as area or total change, over the interval.

Substitution Method

The substitution method simplifies integrals by changing variables to transform a complex integral into a more manageable form. It involves identifying a part of the integrand as a new variable and rewriting the integral in terms of this variable.

Exponential Functions with Variable Exponents

Exponential functions where the exponent is a function of the variable, like 10^{√x}, require careful handling during integration. Understanding how to differentiate and integrate such functions often involves using substitution and properties of logarithms.