37–56. Integrals Evaluate each integral.


∫ eˣ/(36 – e²ˣ), x < ln 6

Verified step by step guidance

1. Identify the integral to solve: \(\int \frac{e^x}{36 - e^{2x}} \, dx\) with the condition \(x < \ln 6\).
2. Recognize that the denominator can be rewritten as a difference of squares: \$36 - e^{2x} = (6)^2 - (e^x)^2$.
3. Use substitution by letting \(u = e^x\). Then, compute \(du = e^x \, dx\), which implies \(dx = \frac{du}{u}\).
4. Rewrite the integral in terms of \(u\): \(\int \frac{u}{36 - u^2} \cdot \frac{du}{u} = \int \frac{1}{36 - u^2} \, du\).
5. Recognize the integral \(\int \frac{1}{a^2 - u^2} \, du\) and recall the formula for this type of integral, which involves partial fractions or inverse hyperbolic functions, then proceed accordingly.

Key Concepts

Integration by Substitution

Integration by substitution is a method used to simplify integrals by changing variables. It involves identifying a part of the integrand as a new variable, which transforms the integral into a simpler form. This technique is especially useful when the integral contains a composite function.

Properties of Exponential Functions

Understanding exponential functions, such as eˣ and e²ˣ, is crucial for manipulating and simplifying expressions within integrals. Knowing how to handle their derivatives and algebraic properties helps in recognizing substitution candidates and simplifying the integrand.

Domain Restrictions and Their Impact

The condition x < ln 6 restricts the domain of the integral, ensuring the denominator 36 – e²ˣ remains positive and the integral is defined. Recognizing domain restrictions helps avoid undefined expressions and guides the choice of substitution and integration limits.