Question

Evaluate the following integrals. Include absolute values only when needed. $$\int \frac{e^{5+\sqrt{x}}}{\sqrt{x}} d x$$

Short answer

Question: Evaluate the integral \(\int \frac{e^{5+\sqrt{x}}}{\sqrt{x}} d x\). Answer: \(\int \frac{e^{5+\sqrt{x}}}{\sqrt{x}} d x = e^{5+\sqrt{x}} + C\)

Step-by-step solution

Find a suitable substitution

Let's make the substitution \(u = 5 + \sqrt{x}\). Now let's find the differential of \(u\) with respect to \(x\). Since \(\frac{du}{dx} = \frac{d}{dx}\left(5 + \sqrt{x} \right)\), we have: $$\frac{d u}{d x} = \frac{1}{2 \sqrt{x}}$$ Now, let's solve for \(dx\): $$d x = 2 \sqrt{x} d u$$

Rewrite the integral in terms of \(u\)

Now, let's rewrite the integral using our substitution \(u = 5 + \sqrt{x}\): $$\int \frac{e^{5+\sqrt{x}}}{\sqrt{x}} d x = \int e^u \left(\frac{2 \sqrt{x}}{2 \sqrt{x}}\right) d u$$ The integral now becomes: $$\int e^u d u$$

Evaluate the integral in terms of \(u\)

The integral of \(e^u\) with respect to \(u\) is simply \(e^u\). Therefore, we have: $$\int e^u d u = e^u + C$$ where \(C\) is the constant of integration.

Substitute back for \(x\)

Now, we need to substitute back the original variable \(x\) in our result. Recall that \(u = 5+\sqrt{x}\): $$e^u + C = e^{5+\sqrt{x}} + C$$

Check if absolute values are needed

In this case, there is no need to include absolute values in the final answer, since the exponential function \(e^{5+\sqrt{x}}\) is always positive.

Final Result

The final result for the given integral is: $$\int \frac{e^{5+\sqrt{x}}}{\sqrt{x}} d x = e^{5+\sqrt{x}} + C$$

Key concepts

Integration by Substitution

Integration by substitution is a fantastic technique used to simplify complex integrals. It works by changing the variable of integration, making the problem easier to handle. The core idea is to substitute a part of the integral with a new variable, which simplifies the integration process.
To use this method effectively, follow these steps:

• Choose a substitution that simplifies your integrand, which is the function you are integrating. For example, in our exercise, we chose u = 5 + \( \sqrt{x} \) to streamline the given expression.
• Use the chain rule for derivatives to express the differential dx in terms of du. In the example, we found that \( dx = 2 \sqrt{x} du \).
• Rewrite the integral in terms of the new variable u. This step often makes the integral simpler, as it converts it into a standard form that's easy to integrate. For instance, the complex integral became \( \int e^u du \).
• Integrate the new, simpler function.
• Once integrated, substitute back the original variable to obtain the answer in terms of the initial variable.

This technique not only simplifies calculations but also enhances problem-solving skills by emphasizing the importance of understanding functions and their behaviors under transformation.

Definite Integrals

Definite integrals are fundamental in calculating the area under a curve within a specific range on the x-axis. They have definite limits, denoted as the lower and upper bounds of integration. Definite integrals provide precise quantities, making them essential in various practical applications, such as calculating distances, areas, and volumes.
To effectively handle definite integrals:

• Identify the integrand and set the upper and lower limits of integration.
• If necessary, apply the integration techniques like substitution to simplify the integral.
• After integrating, evaluate the result at the upper limit and the lower limit, then subtract the lower value from the upper one.

Unlike indefinite integrals, definite integrals do not have a constant of integration because their primary function is to compute a specific value between bounds. They are powerful in solving real-world problems, especially in fields such as physics, engineering, and statistics.

Indefinite Integrals

Indefinite integrals represent a family of functions and are essential in reverse differentiation. They are called indefinite because they lack specific upper and lower limits of integration. Instead, they include a constant of integration, denoted by \( C \), which accounts for the constant difference between any two antiderivatives of a function.
Key points about indefinite integrals include:

• The process of finding an indefinite integral is also known as antiderivation or integration.
• The result of integrating a function \( f(x) \) is another function, \( F(x) + C \), where \( F(x) \) is the antiderivative.
• This constant \( C \) exists because differentiation eliminates constants, making them unidentifiable when reversing the process through integration.

Indefinite integrals are crucial in various mathematical and practical contexts, like solving differential equations and finding general solutions to patterns described by derivatives. Understanding how to manipulate and interpret indefinite integrals is vital for success in calculus and its applications.