Compute Integral Of (28x^6 - 20x^3)e^(4x^7 - 5x^4) From 0 To 1

In the realm of calculus, integrals play a pivotal role in solving a myriad of problems, ranging from finding the area under a curve to determining the volume of a solid. This article delves into the computation of a specific integral, showcasing the techniques and intricacies involved in arriving at an exact solution. The integral in question is:

$\int_0^1(28x^6 - 20x^3)e^{4x^7 - 5x^4} dx$

This integral presents a fascinating challenge due to the presence of the exponential function and the polynomial terms within the integrand. To tackle this, we'll employ a strategic approach involving substitution, a powerful tool that simplifies complex integrals by transforming them into more manageable forms. The key lies in identifying a suitable substitution that unravels the intricate relationship between the terms in the integrand. In this case, we'll focus on the exponent of the exponential function, $4x^7 - 5x^4$, as our candidate for substitution. This choice is guided by the observation that the derivative of this expression, $28x^6 - 20x^3$, appears prominently as a factor in the integrand. This suggests that the substitution will lead to a significant simplification, allowing us to express the integral in terms of a new variable, making it easier to evaluate. As we embark on this journey, we'll not only unravel the solution to this particular integral but also gain insights into the broader principles of integration, enhancing our ability to tackle similar challenges in the future.

To solve the integral, we'll employ the method of substitution, a powerful technique for simplifying integrals. The key to successful substitution lies in identifying a suitable expression within the integrand whose derivative also appears in the integrand. This allows us to transform the integral into a simpler form that we can readily evaluate. In our case, the integral is:

$\int_0^1(28x^6 - 20x^3)e^{4x^7 - 5x^4} dx$

Observing the integrand, we notice that the exponent of the exponential function, $4x^7 - 5x^4$, is a good candidate for substitution. Let's define a new variable, u, as:

$u = 4x^7 - 5x^4$

The rationale behind this choice is that the derivative of u with respect to x, denoted as du/dx, is:

$\frac{du}{dx} = 28x^6 - 20x^3$

Notice that this derivative, $28x^6 - 20x^3$, is precisely the polynomial term that appears as a factor in the original integrand. This is a crucial observation, as it indicates that our substitution will lead to a significant simplification. By substituting u and du/dx, we can rewrite the integral in terms of u, making it easier to evaluate. This strategic choice of substitution is the cornerstone of our solution, as it transforms the integral into a more manageable form. This strategic step sets the stage for the subsequent steps, where we'll rewrite the integral in terms of u, change the limits of integration, and ultimately evaluate the integral to obtain the exact solution.

Having identified the substitution $u = 4x^7 - 5x^4$, our next step is to express the integral entirely in terms of the new variable u. This involves two key transformations: replacing the original variable x with u in the integrand and expressing the differential dx in terms of du. We've already established the relationship between u and x:

$u = 4x^7 - 5x^4$

and we've computed the derivative of u with respect to x:

$\frac{du}{dx} = 28x^6 - 20x^3$

This derivative provides the crucial link between dx and du. We can rearrange the equation to express dx in terms of du:

$du = (28x^6 - 20x^3) dx$

$\implies dx = \frac{du}{28x^6 - 20x^3}$

Now, we can substitute both u and dx into the original integral:

$\int_0^1(28x^6 - 20x^3)e^{4x^7 - 5x^4} dx = \int e^u du$

Notice how the substitution has elegantly simplified the integral. The original integrand, with its complex interplay of polynomial and exponential terms, has been transformed into a simple exponential function in terms of u. This simplification is a testament to the power of substitution as an integration technique. The next step involves addressing the limits of integration. Since we've changed the variable of integration from x to u, we must also transform the limits of integration to reflect this change. This ensures that the integral is evaluated over the correct range in the u-domain, leading us to the accurate solution. This careful transformation of the variable and differential is a cornerstone of the substitution method, allowing us to navigate complex integrals with greater ease.

When we perform a substitution in a definite integral, it's crucial to change the limits of integration to correspond to the new variable. This ensures that we're evaluating the integral over the correct interval in the new variable's domain. In our case, we've substituted $u = 4x^7 - 5x^4$. The original limits of integration were x = 0 and x = 1. We need to find the corresponding values of u for these limits.

When x = 0:

$u = 4(0)^7 - 5(0)^4 = 0$

So, the lower limit of integration in terms of u is 0.

When x = 1:

$u = 4(1)^7 - 5(1)^4 = 4 - 5 = -1$

So, the upper limit of integration in terms of u is -1. Therefore, our integral with the new limits becomes:

$\int_0^1(28x^6 - 20x^3)e^{4x^7 - 5x^4} dx = \int_0^{-1} e^u du$

Notice that the upper limit is now -1, which is less than the lower limit 0. This indicates that we're integrating in the "reverse" direction. To correct this, we can swap the limits of integration and change the sign of the integral:

$\int_0^{-1} e^u du = -\int_{-1}^0 e^u du$

This manipulation ensures that we're integrating in the standard direction, from a lower limit to an upper limit. This adjustment is crucial for obtaining the correct numerical value of the integral. By carefully changing the limits of integration, we've set the stage for the final step: evaluating the integral in terms of u. This transformation of limits is a key aspect of the substitution method, allowing us to seamlessly transition between different variables and maintain the integrity of the integral's value.

Now that we've successfully transformed the integral and adjusted the limits of integration, we're ready to evaluate the integral in terms of the new variable u. Our integral is:

$- \int_{-1}^0 e^u du$

The integral of the exponential function $e^u$ is simply $e^u$. Therefore, we have:

$- \int_{-1}^0 e^u du = -[e^u]_{-1}^0$

To evaluate the definite integral, we substitute the upper and lower limits of integration into the antiderivative and subtract:

$-[e^u]_{-1}^0 = -(e^0 - e^{-1})$

Recall that $e^0 = 1$ and $e^{-1} = \frac{1}{e}$. Substituting these values, we get:

$-(e^0 - e^{-1}) = -(1 - \frac{1}{e})$

Distributing the negative sign, we obtain the final result:

$-(1 - \frac{1}{e}) = -1 + \frac{1}{e}$

Therefore, the value of the integral is:

$\int_0^1(28x^6 - 20x^3)e^{4x^7 - 5x^4} dx = \frac{1}{e} - 1$

This is the exact value of the integral. We've successfully navigated the challenges posed by the integral, employing the method of substitution to simplify the expression and arrive at a precise solution. The result, $\frac{1}{e} - 1$, is a concise and elegant representation of the area under the curve defined by the integrand over the interval [0, 1]. This final step completes our journey, showcasing the power of calculus in solving intricate problems.

In this comprehensive exploration, we successfully computed the definite integral:

$\int_0^1(28x^6 - 20x^3)e^{4x^7 - 5x^4} dx$

Our approach hinged on the strategic use of u-substitution, a powerful technique that simplifies integrals by transforming them into more manageable forms. We identified the exponent of the exponential function, $4x^7 - 5x^4$, as a suitable candidate for substitution, setting $u = 4x^7 - 5x^4$. This choice was guided by the observation that the derivative of u, $28x^6 - 20x^3$, appeared as a factor in the integrand, paving the way for simplification.

By substituting u and du, we transformed the integral into a simpler form: $\int e^u du$. We also meticulously changed the limits of integration to reflect the new variable u, ensuring the accuracy of our result. The original limits, x = 0 and x = 1, were transformed to u = 0 and u = -1, respectively. This led to the integral $- \int_{-1}^0 e^u du$, where we swapped the limits and changed the sign to maintain the correct direction of integration.

Evaluating the integral, we found the antiderivative of $e^u$ to be $e^u$, leading to the expression $-[e^u]_{-1}^0$. Substituting the limits of integration and simplifying, we arrived at the exact value of the integral:

$\int_0^1(28x^6 - 20x^3)e^{4x^7 - 5x^4} dx = \frac{1}{e} - 1$

This solution underscores the effectiveness of u-substitution in tackling complex integrals. The meticulous steps involved, from identifying the appropriate substitution to transforming the limits of integration, highlight the importance of precision and attention to detail in calculus. The final result, $\frac{1}{e} - 1$, stands as a testament to the power of calculus in solving intricate problems and providing exact solutions.