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Chapter 5: Problem 99

In Exercises \(91-108,\) find the indefinite integral. $$\int \frac{e^{2 x}}{1+e^{2 x}} d x$$

Short Answer

Expert verified
The indefinite integral of the function is \[1/2 ln|1 + e^{2x}| + C\].

Step by step solution

01

Identify a Suitable Substitution

A good choice of substitution here is \(u = 1 + e^{2x}\). This simplifies the equation and allows us to rewrite the integral in terms of 'u', which will then allow us to solve it more easily.
02

Compute the Differential of u

Now differentiate the substitution to solve for \(du\). The differential of \(u\) with respect to \(x\) is \(du = 2e^{2x} dx\). You can rearrange this equation to solve for \(dx = du/2e^{2x}\).
03

Substitute terms

Substitute \(u = 1 + e^{2x}\) into the integral and \( dx = du/2e^{2x}\) to get: \[\int \frac{e^{2x}}{u} \cdot \frac{du}{2e^{2x}}\] which simplifies to: \[1/2 \int \frac{1}{u} du\].
04

Solve the Simplified Integral

Observe that the integral is now much simpler and is in a standard form. The infinite integral of \(1/u\) with respect to \(u\) is \(ln|u|\). Multiplying this by the constant \(1/2\) that remains gives: \[1/2 ln|u| + C\].
05

Substitute Back the Original Variable

The last step is to substitute back \(u\) with our original variable \(x\). This gives us the solution in terms of the original variable \(x\): \[1/2 ln|1 + e^{2x}| + C\]. So the integral of the given function is \[1/2 ln|1 + e^{2x}| + C\].

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