Solving Integral 3x² / (4 + X⁶) Dx A Step-by-Step Guide

This article delves into the step-by-step process of evaluating the integral ∫ (3x² / (4 + x⁶)) dx. This type of integral often appears in calculus and requires a clever application of u-substitution to simplify the expression and arrive at the solution. We will break down each step, explaining the reasoning behind the techniques used, making it easier to understand for anyone with a basic knowledge of calculus.

Understanding the Integral

Before we dive into the solution, let's first understand the integral 3x² / (4 + x⁶) dx. Integrals, in calculus, represent the area under a curve. Solving an integral means finding a function whose derivative is the expression inside the integral (the integrand). The integrand in our case is the fraction 3x² / (4 + x⁶). This particular integral looks complex at first glance, but with the right approach, it can be solved relatively easily.

When faced with an integral like this, one common strategy is to look for opportunities to simplify it using techniques like u-substitution. U-substitution is essentially the reverse of the chain rule in differentiation. It allows us to replace a complex expression within the integral with a simpler variable, making the integration process more manageable. The key is to identify a suitable 'u' and its derivative within the integrand. In this case, we'll explore how x³ can be a good candidate for 'u', which will simplify the denominator significantly.

Applying U-Substitution

The key to solving integral 3x² / (4 + x⁶) dx lies in the strategic application of u-substitution. Let's identify a suitable 'u' that simplifies the integral. Observe that the denominator contains x⁶, which can be rewritten as (x³)² . This suggests that substituting u = x³ might be a fruitful approach. The derivative of u with respect to x is du/dx = 3x², which conveniently appears in the numerator of our integrand. This is a strong indication that u-substitution will work well here.

Now, let's formally make the substitution. Let u = x³. Then, du = 3x² dx. Notice how this substitution beautifully simplifies the numerator. We can directly replace 3x² dx in the original integral with du. The denominator, 4 + x⁶, transforms into 4 + (x³)² = 4 + u². Our integral now looks significantly simpler: ∫ (3x² / (4 + x⁶)) dx becomes ∫ (1 / (4 + u²)) du. This transformed integral is much easier to recognize and solve. It resembles the form of a standard integral that we'll discuss in the next section.

The beauty of u-substitution is its ability to transform complex integrals into simpler, recognizable forms. By carefully choosing 'u' and its derivative, we can often manipulate the integral into a form that can be directly integrated using known integration rules and formulas. This technique is a cornerstone of integral calculus and is essential for solving a wide range of integrals. In our case, the substitution u = x³ has successfully simplified the integral, paving the way for the next step in the solution.

Recognizing the Arctangent Form

After performing the u-substitution, our integral now stands as ∫ (1 / (4 + u²)) du. This integral closely resembles a standard integral form, which is the arctangent integral. Recognizing this form is crucial for solving the integral efficiently. The general form of the arctangent integral is ∫ (1 / (a² + x²)) dx = (1/a) arctan(x/a) + C, where 'a' is a constant and 'C' is the constant of integration.

Comparing our integral, ∫ (1 / (4 + u²)) du, with the general form, we can see a clear correspondence. The 'u' in our integral plays the role of 'x' in the general form, and the constant 4 corresponds to a², which means a = 2. Therefore, we can directly apply the arctangent integral formula to solve our integral. It's important to remember this standard integral form as it appears frequently in calculus problems. Recognizing patterns and standard forms is a key skill in mastering integration techniques.

The arctangent function, also known as the inverse tangent function, is a trigonometric function that returns the angle whose tangent is a given number. Its derivative is 1 / (1 + x²), which is why the integral of 1 / (a² + x²) involves the arctangent function. This connection between trigonometric functions and their inverses is fundamental in calculus and allows us to solve integrals that might otherwise seem intractable. By recognizing the arctangent form, we can efficiently solve this part of the integral and move towards the final solution.

Applying the Arctangent Integral Formula

Now that we've recognized the arctangent form, applying the formula is the next step in solving the integral ∫ (1 / (4 + u²)) du. Recall the arctangent integral formula: ∫ (1 / (a² + x²)) dx = (1/a) arctan(x/a) + C. In our case, we have ∫ (1 / (4 + u²)) du, where a² = 4, so a = 2. Substituting these values into the formula, we get:

∫ (1 / (4 + u²)) du = (1/2) arctan(u/2) + C

This is the integral in terms of 'u'. However, we started with an integral in terms of 'x', so we need to substitute back to express our answer in terms of 'x'. Remember that we initially made the substitution u = x³. Replacing 'u' with x³ in our result, we get:

(1/2) arctan(x³/2) + C

This is the final result of the integration. It's crucial to remember the constant of integration, 'C', as it represents the family of functions whose derivative is the integrand. The constant of integration arises because the derivative of a constant is zero, so when we perform indefinite integration, we need to account for the possibility of a constant term in the original function.

Applying the arctangent integral formula is a straightforward process once you've identified the correct form. The key is to carefully match the given integral with the standard form and correctly substitute the values. Don't forget to substitute back to the original variable and include the constant of integration. This step completes the integration process and provides us with the solution to the integral in terms of the original variable.

Substituting Back to x

After applying the arctangent integral formula, we obtained the result (1/2) arctan(u/2) + C. However, this result is in terms of the variable 'u', and our original integral was in terms of 'x'. Therefore, substituting back to x is a crucial step to express the final answer in the correct form. This step ensures that the solution is consistent with the original problem and can be easily interpreted in the context of the given variables.

Recall that we made the substitution u = x³ earlier in the process. To substitute back, we simply replace 'u' with x³ in the expression (1/2) arctan(u/2) + C. This gives us:

(1/2) arctan(x³/2) + C

This is the final form of the integral, expressed in terms of 'x'. It represents the antiderivative of the original integrand, 3x² / (4 + x⁶). The constant of integration, 'C', is still included to represent the family of functions that have the same derivative.

The importance of substituting back to the original variable cannot be overstated. It's a common mistake to forget this step, especially in more complex integration problems involving multiple substitutions. Always remember to express your final answer in terms of the variables that were present in the original problem statement. This ensures that your solution is complete and accurate. In our case, substituting back to 'x' gives us the final solution to the integral, which is (1/2) arctan(x³/2) + C.

The Final Solution

After carefully applying u-substitution, recognizing the arctangent form, and substituting back to the original variable, we have arrived at the final solution for the integral of 3x² / (4 + x⁶) dx. The solution is:

∫ (3x² / (4 + x⁶)) dx = (1/2) arctan(x³/2) + C

This solution represents the family of functions whose derivative is the integrand, 3x² / (4 + x⁶). The constant of integration, 'C', accounts for the fact that the derivative of a constant is zero, so there are infinitely many functions that could have the same derivative.

This final solution demonstrates the power of u-substitution and the importance of recognizing standard integral forms. By strategically choosing a substitution and applying known integration rules, we can solve complex integrals that might initially seem daunting. The arctangent function plays a crucial role in this solution, highlighting the connection between trigonometric functions and their inverses in calculus.

In summary, evaluating the integral ∫ (3x² / (4 + x⁶)) dx involves a series of steps: first, we identify a suitable u-substitution (u = x³). Then, we rewrite the integral in terms of 'u' and recognize the arctangent form. We apply the arctangent integral formula and finally substitute back to 'x' to obtain the final solution. This process showcases the elegance and effectiveness of calculus techniques in solving integration problems.

Conclusion

In conclusion, we have successfully evaluated the integral ∫ (3x² / (4 + x⁶)) dx using a combination of u-substitution and the arctangent integral formula. The process involved identifying a suitable substitution, transforming the integral into a recognizable form, applying the appropriate integration rule, and substituting back to the original variable. The final solution, (1/2) arctan(x³/2) + C, represents the antiderivative of the given integrand.

This example highlights the importance of mastering various integration techniques, particularly u-substitution, and recognizing standard integral forms. These skills are essential for solving a wide range of calculus problems. By carefully analyzing the integrand and choosing the right approach, we can often simplify complex integrals and arrive at elegant solutions. Understanding the underlying principles and practicing these techniques will greatly enhance your problem-solving abilities in calculus.

The ability to solve integrals like this is fundamental in many areas of mathematics, physics, and engineering. Integrals are used to calculate areas, volumes, probabilities, and many other important quantities. Therefore, mastering integral calculus is a valuable skill for anyone pursuing a career in these fields. This detailed walkthrough of the solution provides a solid foundation for tackling similar integration problems in the future. Remember to practice regularly and familiarize yourself with different integration techniques to become proficient in this essential area of calculus.