Evaluate the following integrals.
∫ e³ˣ/(eˣ - 1) dx

Guida verificata passo dopo passo

Step 1: Recognize that the integral involves a fraction with exponential functions. To simplify, consider substituting a variable to make the integral more manageable. Let u = eˣ, which implies that du = eˣ dx.

Step 2: Rewrite the integral in terms of u. Since eˣ = u, the numerator becomes u³, and the denominator becomes u - 1. The dx term is replaced by du/u based on the substitution.

Step 3: The integral now becomes ∫ u³ / (u(u - 1)) du. Simplify the fraction to ∫ u² / (u - 1) du.

Step 4: To solve ∫ u² / (u - 1) du, consider polynomial long division or partial fraction decomposition to break the integrand into simpler terms. Perform the division or decomposition to express the integrand as a sum of simpler fractions.

Step 5: Integrate each term resulting from the decomposition or division separately. Use standard integration techniques, such as the natural logarithm for terms like 1/(u - 1) and power rule for polynomial terms. Finally, substitute back u = eˣ to express the solution in terms of x.

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