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The substitution method is an essential technique for solving integrals by simplifying them into a more manageable form. It involves choosing a new variable, usually denoted as \( u \), to replace a complicated part of the integral. In the given problem, we started with the substitution \( u = e^t \). This choice was logical because it turned the complex exponent into a simple linear expression in \( u \). This approach helps in converting the integral into something simpler, as demonstrated by changing the integral limits when \( t \) approaches \( \ln(3/4) \) and \( \ln(4/3) \). The limits of integration were updated to \( u_1 = 3/4 \) and \( u_2 = 4/3 \), which makes it easier to evaluate the transformed integral at these new bounds. When applying the substitution method, ensure that the differential \( dt \) is also changed to match the new variable \( du = e^t \, dt \), ensuring that the entire integral is expressed in terms of \( u \). This consistent application transforms the entire original expression into a form where conventional integration techniques can be more easily applied.

Trigonometric substitution is a pivotal technique for simplifying integrals involving expressions like \( \sqrt{1+x^2} \). In our example, after initial substitution, the integral became \( \int \frac{du}{(1+u^2)^{3/2}} \). At this point, trigonometric substitution is appropriate. We selected \( u = \tan(\theta) \), which simplifies \( 1 + u^2 \) to \( \sec^2(\theta) \). This dramatically simplifies the integral because it transforms complex algebraic expressions into straightforward trigonometric functions. The derivative \( du \) also transforms using this substitution, as \( du = \sec^2(\theta) \, d\theta \). The manipulated integral then becomes \( \int \cos(\theta) \, d\theta \), which is very straightforward to evaluate. When using trigonometric substitution, it’s crucial to express all parts of the integral, including limits if it’s definite, in terms of trigonometric expressions to facilitate easier integration and ensure we can reverse back correctly to the original variables.

Definite integrals help not only in finding the antiderivative of a function over an interval but also in calculating the accumulated change over that interval. In the given problem, after simplifying with substitutions, we used the definite integral format to find the area under the transformed curve from \( 3/4 \) to \( 4/3 \). Through the solution process, we performed integration over this bounded region, yielding results significant to understanding the area. Finally, after back-substituting to the original variable \( u \), we evaluated the expression \( \frac{u}{\sqrt{1+u^2}} \) at the upper and lower bounds. This evaluation shows the power of definite integrals—providing a precise numerical value or area, as in this problem, ending up with \[ \frac{1}{5} \]. The calculation was straightforward but required careful attention to correctly apply each transformation and substitute back the original variables involving exact fractional arithmetic wherever possible.