7–64. Integration review Evaluate the following integrals.
20. ∫ eˣ (1 + eˣ)⁹ (1 - eˣ) dx

Verified step by step guidance

1. Step 1: Begin by analyzing the integral ∫ eˣ (1 + eˣ)⁹ (1 - eˣ) dx. Notice that the integrand contains terms involving eˣ, (1 + eˣ), and (1 - eˣ). This suggests that substitution might simplify the expression.
2. Step 2: Let u = 1 + eˣ. Then, compute the derivative of u with respect to x: du/dx = eˣ. This implies that du = eˣ dx, which matches the eˣ dx term in the integral.
3. Step 3: Rewrite the integral in terms of u. Substituting u = 1 + eˣ, we also note that (1 - eˣ) can be expressed as (u - 2). The integral becomes ∫ (u⁹)(u - 2) du.
4. Step 4: Expand the integrand. Multiply u⁹ by (u - 2) to get u¹⁰ - 2u⁹. The integral now becomes ∫ (u¹⁰ - 2u⁹) du.
5. Step 5: Integrate term by term. Use the power rule for integration: ∫ uⁿ du = uⁿ⁺¹ / (n + 1). Apply this rule to each term in the expanded integrand to find the antiderivative. After integrating, substitute back u = 1 + eˣ to express the solution in terms of x.

Key Concepts

Integration Techniques

Integration techniques are methods used to evaluate integrals, which can include substitution, integration by parts, and partial fractions. In this case, recognizing the structure of the integrand, particularly the presence of the exponential function and polynomial terms, is crucial for selecting the appropriate technique to simplify the integral.

Substitution Method

The substitution method is a powerful technique in integration that involves changing the variable of integration to simplify the integral. For the given integral, a suitable substitution could involve letting u = eˣ, which transforms the integral into a more manageable form, allowing for easier evaluation.

Definite vs. Indefinite Integrals

Understanding the difference between definite and indefinite integrals is essential in calculus. An indefinite integral, like the one presented, represents a family of functions and includes a constant of integration. Recognizing this helps in interpreting the result of the integral correctly and applying it in further calculations or contexts.