Question

Preliminary steps The following integrals require a preliminary step such as a change of variables before using the method of partial fractions. Evaluate these integrals. $$\int \frac{\left(e^{3 x}+e^{2 x}+e^{x}\right)}{\left(e^{2 x}+1\right)^{2}} d x$$

Short answer

Question: Determine the antiderivative of the given integral: $$\int \frac{e^{3x} + e^{2x} + e^x}{(e^{2x}+1)^2} dx$$ Answer: The antiderivative of the given integral is: $$\frac{1}{2} \ln{(e^{2x}+1)} + \frac{1}{2} \left( \arctan(e^x) + \frac{e^x}{e^{2x}+1}\right) + C$$

Step-by-step solution

Change of Variables

Let's perform a change of variables. Define \(u = e^x\). We'll also need to find the differential \(du\). Since \(u = e^x\), taking the derivative with respect to \(x\) gives us: \(\frac{du}{dx} = e^x\). Now multiply by \(dx\) to get \(du = e^x dx\). Now our integral becomes: $$\int \frac{u^3+u^2+u}{(u^2+1)^2} \cdot \frac{du}{u}$$

Simplify the Integral

Rewrite the integral as: $$\int \frac{u^3+u^2+u}{(u^2+1)^2} du$$

Perform Partial Fraction Decomposition

We want to rewrite the integrand as a sum of simpler fractions. To do this, let's first perform long division of the numerator by the denominator to simplify the expression: The integral becomes: $$\int \frac{u}{u^2+1} du - \int \frac{1}{(u^2+1)^2} du$$

Evaluate the First Integral

The first integral \(\int \frac{u}{u^2+1} du\) can be solved with direct integration: Let \(v = u^2 + 1 \Rightarrow dv = 2u du\). Now substitute \(v\) and \(dv\) into the integral: $$\frac{1}{2} \int \frac{dv}{v}$$ Now we can integrate directly: $$\frac{1}{2} \ln{|v|} + C$$ Substitute back for \(u\): $$\frac{1}{2} \ln{(u^2+1)} + C$$

Evaluate the Second Integral

The second integral is more challenging: $$\int \frac{1}{(u^2+1)^2} du$$ We use the trigonometric substitution \(u = \tan(\theta)\): $$du = \sec^2(\theta)d\theta$$ $$(u^2+1)^2 = (\tan^2(\theta)+1)^2 = (\sec^2(\theta))^2 = \sec^4(\theta)$$ The integral becomes: $$\int \frac{\sec^2(\theta) d\theta}{\sec^4(\theta)}$$ Simplify and integrate: $$\int \frac{d\theta}{\sec^2(\theta)}$$ $$\int \cos^2(\theta) d\theta$$ $$\int \frac{1+\cos(2\theta)}{2} d\theta$$ Now integrate: $$\frac{1}{2} \int (1 + \cos(2\theta))d\theta$$ $$\frac{1}{2} (\theta + \frac{1}{2}\sin(2\theta)) + C$$ Now, convert back to the original variable, \(u\): $$\theta = \arctan(u)$$ $$\sin(2\theta) = \frac{2 \tan(\theta)}{1+\tan^2(\theta)} = \frac{2u}{u^2+1}$$ Substitute back in: $$\frac{1}{2} \left( \arctan(u) + \frac{u}{u^2+1}\right) + C$$

Combine the Two Integrals

Now we combine our results from Steps 4 and 5: $$\frac{1}{2} \ln{(u^2+1)} + \frac{1}{2} \left( \arctan(u) + \frac{u}{u^2+1}\right) + C$$

Change Back to Original Variable

Finally, substitute back \(x\) for \(u\), given that \(u = e^x\): $$\frac{1}{2} \ln{(e^{2x}+1)} + \frac{1}{2} \left( \arctan(e^x) + \frac{e^x}{e^{2x}+1}\right) + C$$ This is the antiderivative of the given integral.

Key concepts

Change of Variables

In integral calculus, a change of variables is a technique used to simplify the integration process. By substituting a new variable in place of an old one, we can transform a complex integral into a simpler one. This method is particularly helpful when dealing with functions that are difficult to integrate directly.

To apply a change of variables, you define a new variable, usually denoted by a symbol like \( u \), and express the original variable in terms of this new one. For instance, if \( u = e^x \), then the derivative \( \frac{du}{dx} = e^x \) allows you to replace \( dx \) with \( \frac{du}{e^x} \).

When you substitute these terms into the integral, you must also convert every instance of the old variable to ensure consistency. This method often reveals underlying simplifications that can make the integration more straightforward.

Trigonometric Substitution

Trigonometric substitution is a powerful technique for evaluating integrals that involve square roots or quadratic expressions. By substituting trigonometric identities, such as \( u = \tan(\theta) \), we transform the integral into a form that's generally easier to deal with.

For example, the expression \( u^2 + 1 \) can be replaced by \( \sec^2(\theta) \), based on the identity \( 1 + \tan^2(\theta) = \sec^2(\theta) \). With this substitution, the integral can often be transformed into a simple trigonometric integral that can be managed with basic integration techniques.

Once the substitution is made, it’s crucial to simplify the integral as much as possible before proceeding with integration. After evaluating the integral, remember to switch back to the original variable using inverse trigonometric functions.

Integral Calculus

Integral calculus focuses on the concept of integration, which is essentially the inverse operation of differentiation. In mathematics, integration is a way to find the area under curves or the total accumulation of quantities.

When solving integrals, particularly indefinite integrals, you aim to find the antiderivative of the function. Techniques such as partial fraction decomposition, trigonometric substitution, and change of variables are often employed to tackle complex integrals.

The definite integral, on the other hand, deals with finding the exact area under a curve over a specified interval. Whether you're evaluating an indefinite or definite integral, understanding the fundamental theorem of calculus is essential, as it connects differentiation with integration.