Problem 40 - College Algebra and Calculus: An Applied Approach

Question

Use any basic integration formula or formulas to find the indefinite integral. State which integration formula(s) you used to find the integral. $$ \int\left(6 x+e^{x}\right) \sqrt{3 x^{2}+e^{x}} d x $$

Step-by-Step Solution

Answer

The indefinite integral of \(\int(6 x+e^{x}) \sqrt{3 x^{2}+e^{x}} dx\) is \(\frac{2}{3}(3x^{2} + e^{x})^\frac{3}{2} + C\).

Step 1: Recognize the Integration by Substitution

One method to solve integrals such as this is the substitution method. This problem requires the substitution \( u = 3 x^{2} + e^{x} \). Thus, its derivative would be \( du = (6 x + e^{x})dx \). Notice that, \( (6 x + e^{x})dx \) is exactly the non-radical part of the original integral.

Step 2: Employ the Substitution

Substitute \( u = 3 x^{2} + e^{x} \) and \( du = (6 x + e^{x})dx \) into the original integral. Due to those substitutions, the integral transforms into \( \int \sqrt{u} du \).

Step 3: Carry out the Integration

The integral now appears as \( \int u^{\frac{1}{2}} du \), which can be solved as \( \frac{2}{3} u^{\frac{3}{2}} + C \), where 'C' stands for the constant of integration. This is due to the power rule for integration, where the power is increased by 1 and the new power is then used as the divisor.

Step 4: Substitute 'u' back

The last step is to transform the integral back to the original variables. This is done by substituting the original form of 'u' back into the integral. Hence, \( u^{\frac{3}{2}} \) is replaced by \( (3x^{2} + e^{x})^\frac{3}{2} \), and the final result is \( \frac{2}{3}(3x^{2} + e^{x})^\frac{3}{2} + C \).

Key Concepts

Integration by Substitution

Integration by substitution, often known as \(u\)-substitution, is a strategic method used in calculus for finding integrals. It's particularly useful when dealing with composite functions. The basic idea behind substitution is to simplify an integral by changing its variable. For example, if you have an integral with some expression inside another expression, substitution might help to simplify it.

To successfully apply substitution, you need to identify a part of the integral as \(u\) (the substitution variable), such that its derivative \(du\) is also present in the integral. This allows you to simplify the integral into a more straightforward form. In our case, we took \(u = 3x^2 + e^x\), making \(du = (6x + e^x)dx\), which was convenient because \((6x + e^x)dx\) matched part of our original expression. With that, we transformed our initial integral into a simpler one, \( int \sqrt{u} \, du \). This new form was much easier to integrate.

Power Rule for Integration

The power rule for integration is a fundamental concept that comes in handy when you deal with integrals of polynomial functions and any functions that can be rewritten to fit this form. The rule states that if you have an integral of the form \( int x^n \, dx \), you add 1 to the exponent and then divide by this new exponent. Mathematically, it's expressed as: \( int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \(C\) is the constant of integration.

In our problem, after substitution, the integral transformed into \( int u^{\frac{1}{2}} \, du \). By applying the power rule here, we added 1 to \(\frac{1}{2}\), which brought our exponent to \(\frac{3}{2}\). We then divided \(u^{\frac{3}{2}}\) by \(\frac{3}{2}\), resulting in \(\frac{2}{3} u^{\frac{3}{2}} + C\). It's important to always remember to add the constant of integration \(C\), as indefinite integrals have this arbitrary constant present.

Basic Integration Formulas

Basic integration formulas are the foundation for solving integrals and involve known rules and identities which simplify the integration process. These include fundamental integrals such as integrals of a constant, power rule, or exponential functions. In our exercise, we utilized these fundamental principles to turn a complex integral into a more manageable form.

In this particular problem, we encountered an expression involving a square root. Often, by knowing basic integrals like \(\int x^n \, dx\) or \(\int e^x \, dx\), you can break down complicated integrals. Identifying these forms allows for immediate application of known results, which simplifies the calculation process. Using the basic integration formula rules effectively involves recognizing the structure of your integral and deciding the best approach or rule for solving it smoothly.