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Evaluate the integral. $$ \int \frac{\left(e^{2 x}+e^{3 x}\right)^{2}}{e^{5 x}} d x $$

   Evaluate the integral.
$$
\int \frac{\left(e^{2 x}+e^{3 x}\right)^{2}}{e^{5 x}} d x
$$

Calculus

Earl W. Swokowski 5th Edition

Chapter 7, Problem 54

Instant Answer

Solved by Expert Steven Clarke

03/15/2022

Step 1

This gives us: $$ \int \frac{e^{4x} + 2e^{5x} + e^{6x}}{e^{5x}} dx $$ Show more…

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Evaluate the integral. $$ \int \frac{\left(e^{2 x}+e^{3 x}\right)^{2}}{e^{5 x}} d x $$

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Key Concepts

Properties of Exponents

Exponential functions often need to be manipulated using the laws of exponents, such as e^(a)/e^(b) = e^(a-b). In this context, applying these properties simplifies the integrand, making the integration process easier by reducing complex exponential expressions to simpler forms.

Polynomial Expansion

The process of squaring a sum, such as (A + B)², requires using the binomial expansion formula to express the square as A² + 2AB + B². Expanding the integrand allows each term to be handled separately, facilitating term-by-term integration.

Integration of Exponential Functions

Once the integrand is expressed as a sum of exponential terms, the integral can be approached by integrating each exponential function individually. This uses the standard integral formula for e^(kx), which is ? e^(kx) dx = (1/k)e^(kx) + C, making the integration process straightforward.

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