Question

Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals. $$\int_{0}^{2} \frac{2 x}{\left(x^{2}+1\right)^{2}} d x$$

Short answer

Question: Evaluate the definite integral: $$\int_{0}^{2} \frac{2x}{(x^2+1)^2} dx$$ Answer: The value of the definite integral is $$\frac{4}{5}$$.

Step-by-step solution

Choose a substitution

Let's choose a substitution to simplify the expression inside the integral. Here, we can use the substitution $$u = x^2 + 1.$$ Now, we need to find its derivative with respect to x: $$\frac{du}{dx} = 2x.$$

Rewrite the integral using the substitution

Now, express the function inside the integral in terms of u and du: $$\int \frac{2x}{(x^2+1)^2} dx = \int \frac{1}{u^2} du$$ To proceed, we also need to change the limits of integration. Currently, the limits are in terms of x, and we need them in terms of u. To do this, use the substitution we chose: When $$x = 0$$, $$u = 0^2 + 1 = 1$$ When $$x = 2$$, $$u = 2^2 + 1 = 5$$ Now, the integral becomes: $$\int_{1}^{5} \frac{1}{u^2} du$$

Evaluate the integral

Now, we can evaluate the integral using the power rule and simplifying: $$\begin{aligned} \int_{1}^{5} \frac{1}{u^2} du &= \int_{1}^{5} u^{-2} du \\ &= \left[-u^{-1}\right]_{1}^{5} \\ &= \left[-\frac{1}{u}\right]_{1}^{5} \end{aligned}$$

Substitute the limits of integration

Finally, apply the limits of integration to find the value of the definite integral: $$\begin{aligned} -\frac{1}{u}\bigg|_{1}^{5} &= -\left(\frac{1}{5} - \frac{1}{1}\right)\\ &= -(-\frac{4}{5}) \\ &= \frac{4}{5} \end{aligned}$$ Therefore, the value of the definite integral is $$\frac{4}{5}.$$

Key concepts

U-Substitution

The technique of u-substitution is a fundamental method in calculus to simplify integrals, and it’s very similar to applying a change of variable in algebra. When you have an integral that’s difficult to solve directly, u-substitution helps you to transform it into a more manageable form.

In essence, u-substitution involves choosing a new variable, u, which is a function of x, usually picking part of the integrand that is more complex. Then, by finding du (the differential of u), you rewrite the original integral in terms of u and du. This step makes the integrand simpler and often familiar to integrate.

The challenge in u-substitution is choosing the right function of x to be u and correctly computing the differential du. It’s important to remember that du must account for the derivative of the chosen u with respect to x, ensuring that every x-term in the integral can be converted into a u-term.

In the given exercise, the substitution u = x² + 1 simplifies the original integral significantly, transforming a complex rational expression into a basic power of u that is much easier to handle.

Integration Limits

When dealing with definite integrals, we encounter integration limits, which specify the interval over which we’re integrating. In the process of u-substitution, it’s critical to adjust these limits to match the new variable u. If this step is skipped, we'd integrate over the wrong interval, leading to an incorrect answer.

Changing the integration limits is straightforward: simply substitute the original x-values into the substitution equation we’ve developed. This gives us the new u-values that correspond to the original x-values at the lower and upper limits of the integral.

In our example, we start with the x-limits of 0 and 2. After the substitution to u, the limits transform into 1 and 5, respectively. As a result, the integration is carried out over the new interval from u=1 to u=5. Remember, this step ensures that the newly formed u-integral evaluates the function over the same portion of the domain as the original x-integral.

Power Rule for Integration

The power rule for integration is a basic yet incredibly useful tool in finding antiderivatives. It states that the integral of xⁿ, where n is any real number other than -1, is ((xⁿ⁺¹)/(n+1)) plus a constant of integration. This rule allows us to quickly find the antiderivative of any power of x.

In our exercise, after applying u-substitution, we have to integrate u to the power of -2. Using the power rule, we add 1 to the exponent and then divide by the new exponent to get the antiderivative, which simplifies down to -u^-1 or -1/u. By applying this technique, the integration process becomes a simple task of manipulating exponents, followed by substituting the integration limits to find the definite integral’s value.