integral of x^4 e^x dx from 0 to 1

 Integration is a fundamental concept in mathematics that finds its applications in various fields, ranging from physics and engineering to economics and biology. It provides a powerful tool to calculate areas, volumes, and quantities that accumulate over time. In this article, we will delve into a specific integral,

${\int }_0^1{�}^4{�}^{�}��$, exploring the techniques and implications of solving it.

Understanding the Integral:

The integral ${\int }_0^1{�}^4{�}^{�}��$ represents the area under the curve of the function ${�}^4{�}^{�}$ from $�=0$ to $�=1$. This seemingly complex function combines a polynomial term (${�}^4$) with an exponential term (${�}^{�}$), making it a challenging yet intriguing problem to solve.

Integration Techniques:

To solve this integral, we can use integration by parts, a method derived from the product rule for differentiation. Integration by parts states:

$$\int ���=��-\int ���$$

For our integral, let's choose $�={�}^4$ and $��={�}^{�}��$. Calculating $��$ and $�$, we get:

$$��=4{�}^3��\text{and}�=\int {�}^{�}��={�}^{�}$$

Applying the integration by parts formula, we have:

$$\int {�}^4{�}^{�}��={�}^4{�}^{�}-\int 4{�}^3{�}^{�}��$$

Now, we need to integrate $\int 4{�}^3{�}^{�}��$. Repeating the integration by parts method, we choose $�=4{�}^3$ and $��={�}^{�}��$, leading to:

$$\int 4{�}^3{�}^{�}��=4{�}^3{�}^{�}-\int 12{�}^2{�}^{�}��$$

We can continue this process iteratively until we obtain a simpler integral that can be easily calculated.

Evaluating the Definite Integral:

To evaluate ${\int }_0^1{�}^4{�}^{�}��$, we substitute the upper limit ($�=1$) and lower limit ($�=0$) into our antiderivative:

$${\int }_0^1{�}^4{�}^{�}��={[{�}^4{�}^{�}-4{�}^3{�}^{�}+12{�}^2{�}^{�}-24�{�}^{�}+24{�}^{�}]}_0^1$$

Evaluating this expression at $�=1$ and $�=0$, we obtain the result of the definite integral.

Applications and Significance:

This integral appears in various real-world problems, especially those involving growth and accumulation. In finance, it can model the compound interest earned on an investment over time. In biology, it might describe the rate of population growth where both the current population and the population growth rate are considered.

Moreover, understanding integrals like these is crucial in probability theory. The integral represents the moment-generating function of a specific probability distribution. Moments provide essential information about the shape and characteristics of a distribution, making them fundamental in statistical analysis.

Conclusion:

In this exploration of the integral ${\int }_0^1{�}^4{�}^{�}��$, we've seen how integration by parts, a powerful technique in calculus, can be applied iteratively to solve complex integrals. Integrals like these find applications in a myriad of fields, emphasizing their significance in practical scenarios. As we continue to advance our knowledge in mathematics, these techniques remain invaluable tools in solving real-world problems, bridging the gap between theory and application.