Question

In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.∫ $$(e^{t} dt) / ((1$$ + $$e^{2t}$)^{3/2}) $$from ln(3/4) to ln(4/3)

Accepted Answer

Start by identifying a substitution to simplify the integral. Notice the expression inside the denominator: $$1 + e$$^{2t}$$. Let’s set x$$ = $$e^{t}. $$This substitution will help rewrite the integral in terms of $$x. $$Since $$x = e$$^{t}$$, then dx$$ = $$e^{t} dt = x dt$, which implies dt$$ = \frac{dx}{x}$$. Also, update the limits of integration: when t$$ = \ln(\frac{3}{4})$$, x$$ = $$e^{\ln(\frac{3}$${4})} = \frac{3}{4}$$; when t$$ = \ln(\frac{4}{3})$$, x$$ = $$e^{\ln(\frac{4}$${3})} = \frac{4}{3}$$. Rewrite the integral in terms of x$: the numerator e$$^{t}$$ dt$$ becomes $$x dt = dx$$, and the denominator becomes $$(1 + x$$^{2}$$)^{3/2}. So $$the integral is now $$\int_{3/4}^{4/3} \frac{dx}{(1 + x$$^{2}$$)^{3/2}$$}$$. Next, use a trigonometric substitution to handle the integral involving 1$$ + $$x^{2}. $$Since $$1 + x$$^{2}$$ resembles the identity 1$$ + \$$tan^{2}$$(	heta) = \$$sec^{2}$$(	heta)$$, let’s set x$$ = 	an(	heta)$$. Then, dx$$ = \$$sec^{2}$$(	heta) d	heta$$. Substitute into the integral: the denominator becomes (1$$ + \$$tan^{2}$$(\$$theta))^{3/2}$$ = (\$$sec^{2}$$(\$$theta))^{3/2}$$ = \$$sec^{3}$$(	heta)$$, and the numerator dx$$ = \$$sec^{2}$$(	heta) d	heta$$. The integral simplifies to $$\int \frac{\$$sec^{2}$$(	heta) d	heta}{\$$sec^{3}$$(	heta)} = \int \cos(	heta) d	heta$$. Also, update the limits of integration by converting x$$ = 	an(	heta)$$ back to $$	heta$$ using $$	heta = \arctan(x)$$.

In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.
∫ (e^{t} dt) / ((1 + e^{2t})^{3/2}) from ln(3/4) to ln(4/3)

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Key Concepts

Substitution Method in Integration

Trigonometric Substitution

Definite Integrals and Limits of Integration

In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.∫ (e^{t} dt) / ((1 + e^{2t})^{3/2}) from ln(3/4) to ln(4/3)