Question

Use formal substitution (as illustrated in Examples 5 and 6 ) to find the indefinite integral. $$ \int \frac{x^{2}+1}{\sqrt{x^{3}+3 x+4}} d x $$

Short answer

The indefinite integral of \(\int \frac{x^{2}+1}{\sqrt{x^{3}+3 x+4}} d x\) is \(x^{3}+3x+4 + C\).

Step-by-step solution

Select the substitution

First, decide on which part of the function to make a substitution. A typical choice is a part of the function whose derivative is another part of the function. Considering this, choose \(u=x^{3}+3x+4\). This simplifies the integral, as the derivative of \(u\) will contain \(x^{2}\), which is present in the numerator.

Compute the derivative of the substitution

Now, calculate the derivative of the substitution \(u=x^{3}+3x+4\). This gives \(\frac{du}{dx}=3x^{2}+3\). Organize it in a way to isolate \(dx\), which gives \(dx=\frac{1}{3x^{2}+3}du\).

Substitute into the integral

Substitute \(u\) and \(dx\) into the integral. The integral becomes \(\int \frac{x^{2}+1}{\sqrt{u}}.\frac{1}{3x^{2}+3}du\). Seeing that \(x^{2} = \frac{3}{3}x^{2}\), this integral can be rewritten as \(\int \frac{3x^{2}+3}{\sqrt{u}}.\frac{1}{3x^{2}+3}du\). Factor out things that cancel out to get \(\int \frac{\sqrt{u}}{\sqrt{u}}du= \int du\).

Compute the integral

The antiderivative of \(du\) is simply \(u\), so compute the integral to get the antiderivative \(\int du = u + C\).

Substitute back the original variables

Finally substitute back the initial expression for \(u\), which was a function of \(x\). This yields \(u + C = x^{3}+3x+4 + C\).

Key concepts

Formal Substitution

In the context of indefinite integrals, formal substitution is a strategy used to simplify an integral, making it easier to solve. This technique involves replacing a part of the integrand with a new variable, traditionally denoted by 'u'. The purpose is to transform a complex expression into a simpler one that is more manageable to integrate.

When applying formal substitution, the first step is to identify a function inside the integrand that, when differentiated, resembles another part of the integrand. By substituting the identified function with 'u', and replacing 'dx' with 'du' according to the derivative of 'u', the integral transforms into an easier problem. For instance, in the exercise provided, selecting \(u = x^3 + 3x + 4\) simplifies the integral and its computation.

Integration Techniques

Integration techniques comprise a variety of mathematical methods designed to find the antiderivative or integral of functions. These techniques include, but are not limited to, substitution, integration by parts, partial fraction decomposition, and trigonometric substitution.

Each technique is suitable for different kinds of functions. For example, substitution is generally used when part of the integrand is the derivative of another part. Integration by parts is useful when the integrand is a product of two functions. Formal substitution is often the first integration technique applied, as it can transform the integrand into a form amenable to further methods.

Antiderivative Computation

An antiderivative of a function is another function whose derivative gives the original function. Computation of antiderivatives, also known as indefinite integration, involves reversing the process of differentiation.

To compute an antiderivative, one must be familiar with fundamental derivative forms and rules of integration. The computation is often facilitated by using integration techniques to manipulate the integrand. In our exercise, the antiderivative of \(du\) was straightforward to compute as \(u + C\), where 'C' represents the constant of integration, a critical component in the calculation of indefinite integrals.

U-Substitution

U-substitution is a specific application of the formal substitution technique. It is a powerful tool for finding the antiderivatives of more complex functions. The general idea is to choose a part of the integrand to be 'u', a variable that simplifies the integral upon substitution.

The derivative \(du\) is then expressed in terms of \(dx\), which allows the integral to be rewritten entirely in terms of 'u'. After integrating with respect to 'u', one must remember to substitute back in terms of the original variable. Our exercise effectively demonstrates u-substitution by integrating a function of 'x' through the substitution of \(u = x^3 + 3x + 4\), which simplified the integrand significantly and led to an easy calculation of the antiderivative.