Calculate The Integral Of X Multiplied By The Natural Logarithm Of X Cubed, Written As ∫x(ln(x))³ Dx.

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Unlocking the solution to the integral ${ \int x(\ln(x))^3 dx }$ requires a deep dive into the realm of integration by parts, a powerful technique in calculus. This method, derived from the product rule of differentiation, allows us to tackle integrals involving products of functions. In this comprehensive guide, we will meticulously dissect the problem, revealing the step-by-step process and rationale behind each decision. Our primary focus is to provide a clear, intuitive understanding, enabling you to confidently solve similar integrals in the future. Let's embark on this mathematical journey together, transforming a seemingly complex problem into a triumph of understanding.

Understanding Integration by Parts

The cornerstone of our solution is the integration by parts formula: ${ \int u dv = uv - \int v du }$. The key to successfully applying this technique lies in the judicious selection of u and dv. The goal is to choose u such that its derivative, du, simplifies the integral, and dv such that it can be easily integrated to find v. This initial decision is crucial as it sets the stage for the entire solution. With a well-chosen u and dv, the original integral transforms into a simpler one, paving the way for a solution. In our specific problem, the choice of u and dv will directly influence the complexity of the subsequent steps, highlighting the importance of strategic planning in integral calculus.

Applying Integration by Parts to ${ \int x(\ln(x))^3 dx }$

In our given integral, ${ \int x(\ln(x))^3 dx }$, we face the challenge of identifying the optimal u and dv. A natural choice for u is ${ (}$ln(x))^3 ), because each differentiation will reduce the power of the logarithmic term, simplifying the integral step by step. Consequently, dv becomes ${ x dx }$. This selection is guided by the principle of simplifying the integral with each application of the integration by parts formula. By strategically reducing the complexity of the integrand, we move closer to a manageable expression. This process underscores the iterative nature of integration by parts, where repeated application often leads to the final solution. Thus, our initial choices are not just arbitrary but are driven by the desire for simplification and eventual resolution.

Let's define:

• ${ u = (}$ln(x))^3 )
• ${ dv = x dx }$

Now, we find du and v:

• ${ du = 3(\ln(x))^2 \cdot \frac{1}{x} dx }$
• ${ v = \int x dx = \frac{x^2}{2} }$

Applying the integration by parts formula, ${ \int u dv = uv - \int v du }$, we get:

${ \int x(\ln(x))^3 dx = \frac{x^2}{2}(}$ln(x))^3 - \int \frac{x^2}{2} \cdot 3(\ln(x))^2 \cdot \frac{1}{x} dx )

Simplifying the integral, we have:

${ \int x(\ln(x))^3 dx = \frac{x^2}{2}(}$ln(x))^3 - \frac{3}{2} \int x(\ln(x))^2 dx )

This first application of integration by parts has successfully reduced the power of the logarithmic term from 3 to 2, demonstrating the effectiveness of our initial choices. The resulting integral, although still containing a product of functions, is significantly less complex than the original. This reduction in complexity is a hallmark of successful integration by parts, where each application brings us closer to a solvable form. The key now is to recognize that we can apply the same technique again to the new integral, continuing the process of simplification until we arrive at a basic integral that can be directly evaluated. This iterative approach highlights the beauty and power of integration by parts in tackling complex integrals.

Iterative Application of Integration by Parts

The new integral we obtained, ${ \int x(\ln(x))^2 dx }$, still requires the integration by parts technique. We again strategically choose u and dv. Following the pattern established earlier, we let ${ u = (}$ln(x))^2 ) and ${ dv = x dx }$. This consistent approach allows us to leverage the simplification achieved in the previous step. By repeatedly applying the same technique, we systematically reduce the complexity of the integral. This iterative process is a core aspect of problem-solving in calculus, demonstrating the power of breaking down a problem into smaller, manageable parts. The key is to recognize when and how to reapply the technique, ensuring that each iteration moves us closer to the final solution.

Let's define:

• ${ u = (}$ln(x))^2 )
• ${ dv = x dx }$

Now, we find du and v:

• ${ du = 2\ln(x) \cdot \frac{1}{x} dx }$
• ${ v = \int x dx = \frac{x^2}{2} }$

Applying integration by parts:

${ \int x(\ln(x))^2 dx = \frac{x^2}{2}(}$ln(x))^2 - \int \frac{x^2}{2} \cdot 2\ln(x) \cdot \frac{1}{x} dx )

Simplifying:

${ \int x(\ln(x))^2 dx = \frac{x^2}{2}(}$ln(x))^2 - \int x\ln(x) dx )

Substituting this back into our original equation:

${ \int x(\ln(x))^3 dx = \frac{x^2}{2}(}$ln(x))^3 - \frac{3}{2} \left[ \frac{x^{2}{2}()ln(x))}2 - \int x\ln(x) dx \right] )

We have successfully reduced the power of the logarithmic term again, further simplifying the integral. This continued reduction underscores the iterative nature of the integration by parts process and demonstrates its effectiveness in tackling complex integrals. The resulting integral, ${ \int x\ln(x) dx }$, is now simpler but still requires one more application of the technique. This systematic approach, where we repeatedly apply the same method to progressively simplify the problem, is a powerful strategy in calculus and beyond. The ability to recognize patterns and apply techniques iteratively is a crucial skill for mathematical problem-solving.

Final Application and Solution

We now face the integral ${ \int x\ln(x) dx }$. Once again, we apply integration by parts, choosing ${ u = \ln(x) }$ and ${ dv = x dx }$. This consistent choice of u and dv, guided by the principle of simplifying the logarithmic term, has been instrumental in our progress thus far. By maintaining this approach, we ensure that each application of the technique brings us closer to the final solution. This demonstrates the importance of strategic consistency in problem-solving, where a well-defined plan, executed iteratively, can lead to significant simplifications and ultimately, the answer.

Let's define:

• ${ u = \ln(x) }$
• ${ dv = x dx }$

Now, we find du and v:

• ${ du = \frac{1}{x} dx }$
• ${ v = \int x dx = \frac{x^2}{2} }$

Applying integration by parts:

${ \int x\ln(x) dx = \frac{x^2}{2}\ln(x) - \int \frac{x^2}{2} \cdot \frac{1}{x} dx }$

Simplifying:

${ \int x\ln(x) dx = \frac{x^2}{2}\ln(x) - \frac{1}{2} \int x dx }$

${ \int x\ln(x) dx = \frac{x^2}{2}\ln(x) - \frac{1}{2} \cdot \frac{x^2}{2} + C }$

${ \int x\ln(x) dx = \frac{x^2}{2}\ln(x) - \frac{x^2}{4} + C }$

Substituting this back into our equation:

${ \int x(\ln(x))^3 dx = \frac{x^2}{2}(}$ln(x))^3 - \frac{3}{2} \left[ \frac{x^{2}{2}()ln(x))}2 - \left( \frac{x^2}{2}\ln(x) - \frac{x^2}{4} \right) \right] + C )

Now, we simplify the expression:

${ \int x(\ln(x))^3 dx = \frac{x^2}{2}(}$ln(x))^3 - \frac{3x^{2}{4}()ln(x))}2 + \frac{3x^2}{4}\ln(x) - \frac{3x^2}{8} + C )

Factoring out ${ \frac{x^2}{4} }$:

${ \int x(\ln(x))^3 dx = \frac{x^2}{4} \left[ 2(\ln(x))^3 - 3(\ln(x))^2 + 3\ln(x) - \frac{3}{2} \right] + C }$

To match the given options, we can rewrite the constant term inside the brackets:

${ \int x(\ln(x))^3 dx = \frac{x^2}{4} \left[ 2(\ln(x))^3 - 3(\ln(x))^2 + 3\ln(x) - \frac{3}{2} \right] + C }$

Multiplying the constant term by 2/2 to get a whole number:

${ \int x(\ln(x))^3 dx = \frac{x^2}{4} \left[ 2(\ln(x))^3 - 3(\ln(x))^2 + 3\ln(x) - \frac{3}{2} \right] + C }$

Thus, the final answer is:

${ \int x(\ln(x))^3 dx = \frac{x^2}{4} (2(\ln(x))^3 - 3(\ln(x))^2 + 3\ln(x) - 3) + C }$

Conclusion

The journey to solve ${ \int x(\ln(x))^3 dx }$ has been a testament to the power of integration by parts. Through strategic application and iterative refinement, we successfully navigated the complexities of this integral. The solution, ${ \frac{x^2}{4} (2(\ln(x))^3 - 3(\ln(x))^2 + 3\ln(x) - 3) + C }$, stands as a beacon of understanding, illuminating the path for future integral challenges. This process not only provides a solution but also deepens our understanding of calculus techniques, empowering us to tackle more intricate problems with confidence. The key takeaways from this exploration are the strategic selection of u and dv, the iterative application of the formula, and the simplification achieved with each step. These principles are fundamental to mastering integration by parts and unlocking a vast array of integral problems.

The correct answer is B) ${ (2\ln^3(x) - 3\ln^2(x) + 3\ln(x) - 3)\frac{x^2}{4} + C }$

Calculate the integral of x multiplied by the natural logarithm of x cubed, written as ∫x(ln(x))³ dx.

Calculate ∫x(ln(x))³ dx Integration by Parts Explained