Problem 47 - Applied Calculus

Question

Find the indefinite integrals. $$ \int\left(x+\frac{1}{\sqrt{x}}\right) d x $$

Step-by-Step Solution

Answer

\( \frac{x^2}{2} + 2\sqrt{x} + C \)

Step 1 - Separate the Integral

The integral given is \( \int (x + \frac{1}{\sqrt{x}}) \, dx \). We can separate this into two integrals: \( \int x \, dx + \int \frac{1}{\sqrt{x}} \, dx \). This is possible due to the additive property of integrals.

Step 2 - Integrate the First Term

The first integral is \( \int x \, dx \). The antiderivative of \( x^n \) is \( \frac{x^{n+1}}{n+1} + C \). Since here \( n = 1 \), we get \( \frac{x^{1+1}}{1+1} = \frac{x^2}{2} + C_1 \).

Step 3 - Simplify the Second Term

Rewrite \( \frac{1}{\sqrt{x}} \) as \( x^{-1/2} \). This helps to use the power rule for integration.

Step 4 - Integrate the Second Term

Now integrate \( \int x^{-1/2} \, dx \). Applying the power rule, the antiderivative is \( \frac{x^{(-1/2)+1}}{(-1/2)+1} = \frac{x^{1/2}}{1/2} = 2x^{1/2} + C_2 \).

Step 5 - Combine the Results

Add the results of the two integrals: \( \frac{x^2}{2} + 2x^{1/2} + C \). Here, \( C \) represents the sum of constants of integration \( C_1 + C_2 \).

Step 6 - Finalize the Solution

The indefinite integral \( \int \left(x + \frac{1}{\sqrt{x}}\right) \, dx \) evaluates to \( \frac{x^2}{2} + 2\sqrt{x} + C \). This is the final result of integrating the given function.

Key Concepts

Additive Property of Integrals

In calculus, when you work with indefinite integrals, especially when dealing with sums of functions, the additive property becomes very handy. This property states that the integral of a sum is equal to the sum of the integrals of the individual functions.
For example, if you have the integral \( \int (f(x) + g(x)) \, dx \), it can be split into two separate integrals: \( \int f(x) \, dx + \int g(x) \, dx \). This allows solving long, tedious integrals in smaller, more manageable parts.
This method was applied in our original exercise, where the integral \( \int (x + \frac{1}{\sqrt{x}}) \, dx \) was divided into two separate integrals: \( \int x \, dx \) and \( \int \frac{1}{\sqrt{x}} \, dx \). This separation simplifies the process significantly.

Power Rule for Integration

The power rule for integration is a fundamental tool used when integrating functions of the form \( x^n \). This rule states that the integral of \( x^n \) with respect to \( x \) is \( \frac{x^{n+1}}{n+1} + C \), where \( C \) is the constant of integration.

• It is crucial that \( n eq -1 \), since this would result in division by zero.
• The power rule helps convert a complex expression into a simpler form to integrate.

In our problem, it was used twice. First, to integrate \( \int x \, dx \) where \( n = 1 \), resulting in \( \frac{x^2}{2} + C_1 \). Second, it was applied after rewriting \( \frac{1}{\sqrt{x}} \) as \( x^{-1/2} \), enabling its integration to \( 2x^{1/2} + C_2 \).
Using this rule makes integration more accessible and less daunting.

Antiderivative

Finding the antiderivative of a function is essentially reversing the process of differentiation. It provides functions whose derivative yields the given function. This is particularly useful in determining indefinite integrals, which are antiderivatives with a constant of integration \( C \) added.

• An antiderivative of a given function \( f(x) \) is any function \( F(x) \) such that \( F'(x) = f(x) \).
• Indefinite integrals represent the entire family of antiderivatives of a function.

In the solution of our exercise, each integral led to finding an antiderivative:
- For \( \int x \, dx \), the antiderivative is \( \frac{x^2}{2} \).
- For \( \int \frac{1}{\sqrt{x}} \, dx \) or \( \int x^{-1/2} \, dx \), the antiderivative is \( 2x^{1/2} \).
Thus, the sum of these antiderivatives provides the solution for the indefinite integral with an added constant \( C \).