37–56. Integrals Evaluate each integral.
∫₀⁴ sech²√x / √x dx

Verified step by step guidance

1. Recognize that the integral is \( \int_0^4 \frac{\text{sech}^2(\sqrt{x})}{\sqrt{x}} \, dx \). The integrand involves \( \sqrt{x} \) both inside the hyperbolic secant squared function and in the denominator, suggesting a substitution involving \( \sqrt{x} \).
2. Make the substitution \( t = \sqrt{x} \), which implies \( x = t^2 \). Then, differentiate to find \( dx \) in terms of \( dt \): \( dx = 2t \, dt \).
3. Rewrite the integral in terms of \( t \): replace \( \sqrt{x} \) with \( t \), and \( dx \) with \( 2t \, dt \). The integral becomes \( \int_{t=0}^{t=2} \frac{\text{sech}^2(t)}{t} \cdot 2t \, dt \). Notice that the \( t \) in the denominator and numerator will cancel out.
4. Simplify the integral to \( \int_0^2 2 \text{sech}^2(t) \, dt \). This is now a standard integral involving \( \text{sech}^2(t) \), which is the derivative of \( \tanh(t) \).
5. Integrate \( 2 \text{sech}^2(t) \) with respect to \( t \) to get \( 2 \tanh(t) + C \), then apply the limits from 0 to 2 to express the definite integral.

Key Concepts

Integration by Substitution

Integration by substitution is a technique used to simplify integrals by changing variables. It involves substituting a part of the integral with a new variable to transform the integral into a more manageable form. This method is especially useful when the integral contains composite functions, such as √x inside another function.

Hyperbolic Secant Function (sech)

The hyperbolic secant function, sech(x), is defined as 1/cosh(x), where cosh(x) is the hyperbolic cosine. Its square, sech²(x), often appears in integrals related to hyperbolic functions. Understanding its properties and derivatives helps in recognizing integrals that can be simplified using known formulas.

Definite Integrals and Limits of Integration

Definite integrals calculate the net area under a curve between two specific points, called limits of integration. Evaluating a definite integral involves finding the antiderivative and then applying the Fundamental Theorem of Calculus to compute the difference at the upper and lower limits.