Question

MATH 183-5 5. Evaluate the integral. ∫ 1+8eˣ/1-eˣd x

Answer 1

The integral ∫ (1 + 8e^x) / (1 - e^x) dx requires a substitution method to simplify and solve. The provided excerpts do not directly relate to this integral, and the clarification of appropriate techniques, such as substitution, is necessary.

Step-by-step explanation:

The student is tasked with evaluating an integral of a rational function. The integral is ∫ (1 + 8e^x) / (1 - e^x) dx. To solve this integral, one could attempt various methods such as substitution, partial fractions, or integration by parts, depending on the complexity of the integrand.

However, the provided information appears to be unrelated excerpts from a different context, mentioning procedures like integrating around a constant radius, discussing odd and even functions, and integrating potential energy, which doesn't directly apply to solving the integral in question. The correct approach would require finding a substitution that simplifies the integral, such as setting u = 1 - e^x, and then finding du and substituting back into the integral to solve. The task would conclude by evaluating the new integral in terms of u and then substituting back to find the solution in terms of x.

Answer 2

To evaluate the integral ∫ (1+8eˣ)/(1-eˣ) dx, we can use a substitution. Let u = 1 - eˣ. Then, differentiating both sides with respect to x, du = -eˣ dx. Rearranging, we find that dx = -du/eˣ. Substituting these values into the integral, we have ∫ (1+8eˣ)/(1-eˣ) dx = ∫ (1+8u)/u (-du/eˣ). Simplifying, we get ∫ (1+8u)/u^2 du.

Step-by-step explanation:

To evaluate the integral ∫ (1+8eˣ)/(1-eˣ) dx, we can use a substitution. Let u = 1 - eˣ. Then, differentiating both sides with respect to x, du = -eˣ dx. Rearranging, we find that dx = -du/eˣ. Substituting these values into the integral, we have ∫ (1+8eˣ)/(1-eˣ) dx = ∫ (1+8u)/u (-du/eˣ). Simplifying, we get ∫ (1+8u)/u^2 du.

To solve this integral, we can rewrite it as two separate integrals: ∫ du/u^2 + ∫ 8u/u^2 du. The first integral, ∫ du/u^2, simplifies to -1/u. The second integral, ∫ 8u/u^2 du, simplifies to 8 ∫ du/u. Combining these results, we have -1/u + 8 ln|u| + C, where C is the constant of integration.

Finally, substituting u back in terms of x, we have -1/(1-eˣ) + 8 ln|1-eˣ| + C for the original integral.

The complete question is: MATH 183-5 5. Evaluate the integral. ∫ 1+8eˣ/1-eˣd x is: