- #1
karush
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$$\int_{}^{} \frac{e^{2x}}{e^{2x}-2}dx. \\u=e^{2x}-2\\du=e^{2x}$$
Now what?
Now what?
karush said:$\int_{}^{}\frac{1}{u}\ du\ dx = \ln\left({e^{2x}-2}\right)/2$
An integral is a mathematical concept that represents the area under a curve in a graph. It is a fundamental tool in calculus and is used to calculate a wide range of quantities such as displacement, volume, and probability.
Evaluating an integral allows us to find the exact numerical value of the area under a curve, which may have practical applications in various fields such as physics, engineering, and economics. It also helps us to understand the behavior and properties of a function.
To evaluate an integral, we use a set of rules and techniques such as substitution, integration by parts, and trigonometric identities. The specific method used depends on the form of the integral and the functions involved.
No, not every integral can be evaluated analytically. Some integrals have no closed-form solution and can only be approximated using numerical methods. However, most integrals encountered in basic calculus can be evaluated using the techniques mentioned above.
The "u" in this integral represents the variable with respect to which we are integrating. It is often used in substitution methods to simplify the integral and make it easier to evaluate. In this case, the integral can be rewritten as $$\int e^{2x} \frac{1}{u} du$$, making it easier to integrate using the power rule.