Question

Evaluate the integrals in Exercises 39–54.∫ (e⁴t + 2e²t - e^t) / (e²t + 1) dt

Accepted Answer

Start by examining the integral $$ \int \frac{e^{4t} + 2e^{2t} - e^{t}}{e^{2t} + 1} \, dt . $$Notice that the numerator and denominator involve exponential functions with powers related to $$ t . $$Make a substitution to simplify the expression. Let $$ u = e^{2t} . $$Then, compute $$ du  in $$terms of $$ dt $$: $$ du = 2e^{2t} dt = 2u dt $$, so $$ dt = \frac{du}{2u} . $$Rewrite the integral in terms of $$ u . $$Express each term in the numerator and denominator using $$ u $$: $$ e^{4t} = (e^{2t})^2 = u^2 $$, $$ 2e^{2t} = 2u $$, and $$ e^{t} = e^{t}  ($$which can be expressed as $$ e^{t} = (e^{2t})^{1/2} = u^{1/2} ). $$Substitute these into the integral and replace $$ dt $$ with $$ \frac{du}{2u} . $$Simplify the resulting integral in terms of $$ u $$, combining like terms and simplifying the fraction if possible. This should reduce the integral to a rational function of $$ u $$ and possibly $$ u^{1/2} . $$Once simplified, split the integral into simpler parts if needed and integrate each term with respect to $$ u . $$After integrating, substitute back $$ u = e^{2t}  to $$express the answer in terms of $$ t .$$

Evaluate the integrals in Exercises 39–54.
∫ (e⁴t + 2e²t - e^t) / (e²t + 1) dt

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Key Concepts

Integration of Rational Functions

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Substitution Method in Integration