Question

Evaluate the following integrals. $$\int e^{\sqrt[7]{x}} d x$$

Short answer

Question: Evaluate the integral: $$\int e^{\sqrt[7]{x}} dx$$ Answer: $$\int e^{\sqrt[7]{x}} dx = 7 \sum_{k=1}^{6} (-1)^{k-1} k x^{\frac{6-k}{7}} e^{\sqrt[7]{x}} + C$$

Step-by-step solution

Apply Substitution

Let's substitute \(u = \sqrt[7]{x}\), so our x can be written as \(x = u^7\). To change our dx in terms of du, we need to differentiate x with respect to u: $$\frac{dx}{du} = 7u^6 \Rightarrow dx = 7u^6 du$$ Now, we can write our integral in terms of u: $$\int e^{\sqrt[7]{x}} dx = \int e^u 7u^6 du$$

Find the Antiderivative

Now we will find the antiderivative of the simplified function with respect to u: $$\int e^u 7u^6 du = 7 \int u^6 e^u du$$ To solve this integral, we'll use integration by parts. Let's choose \(dv = e^u du\) and \(u = u^6\). Now we find du and v: $$du = 6u^5 du$$ $$v = \int e^u du = e^u$$ Now we have to apply integration by parts: $$\int u^6 e^u du = u^6 \cdot e^u - \int e^u 6u^5 du = u^6 e^u - 6 \int u^5 e^u du$$ We still have an integral left to solve which involves the product of the exponential and a polynomial function. We can apply integration by parts again: Take \(dv = e^u du\) and \(u = u^5\). Then find du and v: $$du = 5u^4 du$$ $$v = \int e^u du = e^u$$ Apply integration by parts again: $$\int u^5 e^u du = u^5 \cdot e^u - \int e^u 5u^4 du = u^5 e^u - 5 \int u^4 e^u du$$ We can see a pattern emerging: the exponent of u in the integral decreases by 1 in each step, while the constant in front of the integral increases by 1. We can keep applying integration by parts until we reach a point where we need to solve \(\int e^u du\): $$\int e^{\sqrt[7]{x}} dx = 7 \Big[ u^6 e^u - 6 \big( u^5 e^u - 5 \dots -( u e^u - \int e^u du )\dots \big) \Big]$$ The last integral in the sequence is very simple to integrate: $$\int e^u du= e^u+C$$ Now we have the antiderivative in terms of u.

Substitute Back the Original Variable

Now we'll substitute back the original variable x: $$\int e^{\sqrt[7]{x}} dx = 7 \Big[ (\sqrt[7]{x})^6 e^{\sqrt[7]{x}} - 6 \big((\sqrt[7]{x})^5 e^{\sqrt[7]{x}} - 5 \dots -( \sqrt[7]{x} e^{\sqrt[7]{x}} - e^{\sqrt[7]{x}} )\dots \big) \Big]+ C$$ This is the solution for the integral: $$\int e^{\sqrt[7]{x}} dx = 7 \sum_{k=1}^{6} (-1)^{k-1} k x^{\frac{6-k}{7}} e^{\sqrt[7]{x}} + C$$

Key concepts

Substitution Method

The substitution method is a helpful tool in calculus for solving integrals, especially when dealing with complex functions. Essentially, it involves replacing a part of the integral with a new variable to simplify the integral's structure.

In this particular exercise, we begin by substituting \( u = \sqrt[7]{x} \), which converts the original integral into a function of \( u \) instead of \( x \). This makes the integral easier to manage because the power of \( u \) simplifies calculations. By differentiating \( x = u^7 \) with respect to \( u \), we find \( dx = 7u^6 \, du \). This new expression transforms the integral into:

• \( \int e^{u} \cdot 7u^6 \, du \)

This substitution significantly simplifies the integration process and sets the stage for using other techniques like integration by parts.

Exponential Integrals

Exponential integrals can seem daunting but are quite manageable with proper techniques. They are integrals that involve an exponential function of a variable within the integrand. For this exercise, after simplifying the integral through substitution, we focus on the integral of the form \( \int u^n e^u \, du \).

To find solutions to exponential integrals, particularly when combined with polynomial expressions like \( u^n \), integration by parts is an excellent choice. This method allows us to simplify the product of two functions that aren't straightforward to integrate together. Here's how it works:

• Choose which function to differentiate and which to integrate. Generally, the polynomial part \( u^n \) is differentiated, and the exponential part \( e^u \) is integrated.
• Apply the integration by parts formula: \( \int u^n e^u \, du = u^n e^u - \int e^u n \, u^{n-1} \, du \).
• Continue this step iteratively for the resulting integral until \( n = 0 \).

This process yields a solution involving a polynomial decreased by one degree at each step alongside the exponential function until a straightforward exponential integral is reached.

Polynomial Functions

Polynomial functions are mathematical expressions containing variables raised to powers combined via addition, subtraction, and multiplication. In the realm of integrals, they often require special techniques for successful integration, especially when combined with other functions like exponentials.

In our exercise, we encounter a polynomial function through the substitution \( u = \sqrt[7]{x} \), making \( u^6 \), \( u^5 \), etc. during the integration by parts process. Here are key considerations when dealing with polynomial functions within integrals:

• Observe the highest degree of the polynomial, as it determines the number of iterations required when performing techniques like integration by parts.
• Recognize that each iteration simplifies the polynomial by reducing the power by one until reaching a constant term.
• Each step reduces complexity, ultimately leading to the integration of the remaining exponential function, which is straightforward.

By managing polynomials correctly, they transform complex integration problems into manageable solutions, where each decrement of a polynomial's degree brings you closer to the integral's evaluation.