Problem 34 - College Algebra and Calculus: An Applied Approach

Question

Find the indefinite integral and check your result by differentiation. $$ \int\left(\sqrt[4]{x^{3}}+1\right) d x $$

Step-by-Step Solution

Answer

The indefinite integral of \( \sqrt[4]{x^{3}}+1 \) is \( (4/7)x^{7/4} + x + C \).

Step 1: Finding the Integral

We have to integrate the function \( \sqrt[4]{x^{3}} + 1 \). If we write this in more simple terms, it's \( \int (x^{3/4} + 1) \, dx \). Now, applying the rule of integral \( \int x^n \, dx = (1/(n+1)) x^{n+1} + C \), where C is the constant of integration, for both terms: \( \int (x^{3/4} + 1) \, dx = (4/7)x^{7/4} + x + C \)

Step 2: Differentiating to Check the Answer

The next step is to differentiate our result and check if it matches the original function. The derivative of \( (4/7)x^{7/4} + x + C \) is \( (4/7)*(7/4)x^{3/4} + 1 = x^{3/4} + 1 \). As this matches the original function, we can confirm the solution is correct.

Key Concepts

Integral Calculus

Integral calculus is a fundamental part of mathematics that deals with the accumulation of quantities, such as areas under curves and the net change across a range. At its core, it's the reverse process of differentiation, and its goal is to find the original function given its derivative, which is also known as the antiderivative.

When you look at the process of finding the indefinite integral, or antiderivative, it's about piecing together a function that, when differentiated, will give the function you started with. The concrete example given in the exercise, \[\int(\sqrt[4]{x^{3}} + 1) dx\], illustrates finding an indefinite integral. We are searching for a function that, once differentiated, yields \(\sqrt[4]{x^{3}} + 1\).

The notation \(\int f(x) dx\) represents the indefinite integral of a function \(f(x)\) with respect to \(x\). The result of this operation will always include a constant, denoted by \(C\), because differentiation of a constant is zero, and so it doesn't influence the derivative of the resulting function. Therefore, the indefinite integral represents a family of functions that all vary by a constant amount.

Integration by Substitution

Understanding Substitution

In the context of integral calculus, integration by substitution is akin to the algebraic 'change of variable' technique. Substitution can often simplify an integral, making it easier to solve. Essentially, you select a part of the integral's function to replace with a new variable. This method is particularly useful when dealing with composite functions.

In situations where integration by substitution is applicable, we look for a function inside another function—imagine a part of the integrand (the function we are integrating) that is itself a function of another variable. By substituting this part with a new variable \(u\), we can frequently transform a challenging integral into a much simpler form that we know how to integrate.

Although the original exercise \(\int(\sqrt[4]{x^{3}} + 1) dx\) did not require substitution due to its straightforwardness, more complex integrals might need this strategy for an easier solution. Substitution is a powerful tool in your calculus toolbox, especially when you encounter more complicated integrands that do not directly fit the basic forms or rules of integration.

Power Rule for Integration

Simplified Integration

The power rule for integration is a straightforward principle that allows us to integrate any real power of \(x\), provided the power is not equal to -1. According to this rule, the integral of \(x^n\) with respect to \(x\) is \(\int x^n dx = \frac{1}{n+1} x^{n+1} + C\), where \(C\) is the constant of integration.

The rule comes as a natural consequence of the reverse operation of differentiation. When you differentiate \(x^{n+1}\), you bring down the power to multiply in front of \(x\), and reduce the power by one according to the derivative power rule. Therefore, when integrating, we increase the power by one and divide by this new power.

In the given exercise, applying the power rule for integration gives us the term \[\frac{4}{7}x^{7/4}\] as the antiderivative of \(x^{3/4}\), and the term \(x\) as the antiderivative of 1 (since \(x^0 = 1\)). Understanding and memorizing this rule is crucial for students as it's one of the most commonly used techniques in finding antiderivatives for polynomials and other algebraic expressions.