Problem 63 - 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: Distribute the Integral

The given expression is \(\int \left( x + \frac{1}{\sqrt{x}} \right) \, dx\). The first step is to distribute the integral over each term inside the parentheses: \(\int x \, dx + \int \frac{1}{\sqrt{x}} \, dx\). This breaks the problem into two smaller problems.

Step 2: Integrate the First Term

The integral of \(x\) with respect to \(x\) is \(\int x \, dx\). Using the power rule for integration \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\), we substitute \(n = 1\) to get \(\frac{x^{1+1}}{1+1} = \frac{x^2}{2}\).

Step 3: Simplify the Second Term

Rewrite the second term \(\frac{1}{\sqrt{x}}\) as \(x^{-\frac{1}{2}}\). The integral now becomes \(\int x^{-\frac{1}{2}} \, dx\).

Step 4: Integrate the Second Term

Use the power rule to integrate \(x^{-\frac{1}{2}}\), \(\int x^{-\frac{1}{2}} \, dx = \frac{x^{-rac{1}{2}+1}}{-\frac{1}{2}+1} + C\). Simplifying this gives \(\frac{x^{\frac{1}{2}}}{\frac{1}{2}} = 2x^{\frac{1}{2}} = 2\sqrt{x}\).

Step 5: Combine the Results

Add the integrals of the separate terms to find the final indefinite integral:\[\int \left( x + \frac{1}{\sqrt{x}} \right) \, dx = \frac{x^2}{2} + 2\sqrt{x} + C,\]where \(C\) is the constant of integration.

Key Concepts

Integration by Parts

Integration by parts is a useful technique when the product of two functions is being integrated. However, in this particular exercise, the concept does not apply directly. Instead, each term in the expression was integrated separately. Integration by parts can be helpful when dealing with functions that are products of algebraic terms, trigonometric functions, exponential functions, or logarithms. Its formula is:\[\int u \, dv = uv - \int v \, du\]where:- \( u \) is a function chosen for its simplicity when differentiated,- \( dv \) is chosen to simplify the integral after integrating.By choosing suitable \( u \) and \( dv \), we can transform the problem into a simpler one. Although not used in this example, being familiar with this method helps in complex integrals.

Power Rule

The power rule is a fundamental concept in calculus for integrating functions of the form \(x^n\). It's incredibly straightforward and is used extensively in both differentiation and integration. When integrating, we can express it as:\[\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\]where \( n eq -1 \), and \( C \) is the constant of integration.In this problem, the power rule was applied to both terms:- For \( x \), we used \( n=1 \), resulting in \( \frac{x^2}{2} \).- For \( \frac{1}{\sqrt{x}} \), rewritten as \( x^{-\frac{1}{2}} \), we set \( n = -\frac{1}{2} \), resulting in \( 2\sqrt{x} \).This rule is especially powerful for polynomials, allowing quick and easy integration.

Indefinite Integration

Indefinite integration refers to finding a function whose derivative is the given function. It is the reverse process of differentiation. When you compute the indefinite integral of a function \( f(x) \), you find the family of functions \( F(x) \) such that \( F'(x) = f(x) \).Let's break down what indefinite integration involves:- **Antiderivative:** The result of the indefinite integral, which is the original function plus a constant \( C \).- **Constant of Integration (C):** Represents all possibles shifts of the antiderivative along the y-axis, as differentiation of any constant is zero.In the exercise, the indefinite integrals of \( x \) and \( \frac{1}{\sqrt{x}} \) were calculated separately and combined. This process finds a general form of the function before differentiation, crucial for solving real-world problems like calculating areas under curves or solving differential equations.