Find the Antiderivative f(x)=(8+x+x^2)/( square root of x)
The problem asks for the computation of the antiderivative, or the indefinite integral, of the given function f(x) = (8 + x + x^2) / √x. The function expressed as a combination of x and its powers is divided by the square root of x, which can also be written as x raised to the power of 1/2. To find the antiderivative, one would typically look to apply integration techniques such as substitution, partial fractions (if applicable), or algebraic manipulation to simplify the expression before integrating. The goal is to express the result as a function of x plus the constant of integration, often denoted as C.
$f \left(\right. x \left.\right) = \frac{8 + x + x^{2}}{\sqrt{x}}$
Answer
Solution:
Step 1:
Identify the antiderivative $F(x)$ by integrating the function $f(x)$.
$$F(x) = \int f(x) \, dx$$
Step 2:
Write down the integral to be solved.
$$F(x) = \int \frac{8 + x + x^2}{\sqrt{x}} \, dx$$
Step 3:
Express $\sqrt{x}$ as $x^{1/2}$.
$$\int \frac{8 + x + x^2}{x^{1/2}} \, dx$$
Step 4:
Rewrite the integrand by inverting $x^{1/2}$ to $x^{-1/2}$.
$$\int (8 + x + x^2) x^{-1/2} \, dx$$
Step 5:
Apply exponent multiplication rules.
Step 5.1:
Use the rule $(a^m)^n = a^{mn}$.
$$\int (8 + x + x^2) x^{1/2 \cdot -1} \, dx$$
Step 5.2:
Combine the exponents.
$$\int (8 + x + x^2) x^{-1/2} \, dx$$
Step 5.3:
Maintain the negative exponent.
$$\int (8 + x + x^2) x^{-1/2} \, dx$$
Step 6:
Distribute $x^{-1/2}$ across the sum.
Step 6.1:
Apply the distributive property.
$$\int (8x^{-1/2} + x \cdot x^{-1/2} + x^2 \cdot x^{-1/2}) \, dx$$
Step 6.2:
Raise $x$ to the power of 1.
$$\int (8x^{-1/2} + x^1 \cdot x^{-1/2} + x^2 \cdot x^{-1/2}) \, dx$$
Step 6.3:
Combine like bases using the rule $a^m a^n = a^{m+n}$.
$$\int (8x^{-1/2} + x^{1 - 1/2} + x^{2 - 1/2}) \, dx$$
Step 6.4:
Express 1 as a fraction to combine exponents.
$$\int (8x^{-1/2} + x^{2/2 - 1/2} + x^{2 - 1/2}) \, dx$$
Step 6.5:
Combine the numerators.
$$\int (8x^{-1/2} + x^{1/2} + x^{2 - 1/2}) \, dx$$
Step 6.6:
Simplify the exponents.
$$\int (8x^{-1/2} + x^{1/2} + x^{3/2}) \, dx$$
Step 7:
Separate the integral into individual terms.
$$\int 8x^{-1/2} \, dx + \int x^{1/2} \, dx + \int x^{3/2} \, dx$$
Step 8:
Factor out constants from the integrals.
$$8 \int x^{-1/2} \, dx + \int x^{1/2} \, dx + \int x^{3/2} \, dx$$
Step 9:
Integrate $x^{-1/2}$ using the Power Rule.
$$8(2x^{1/2} + C) + \int x^{1/2} \, dx + \int x^{3/2} \, dx$$
Step 10:
Integrate $x^{1/2}$ using the Power Rule.
$$8(2x^{1/2} + C) + \frac{2}{3}x^{3/2} + C + \int x^{3/2} \, dx$$
Step 11:
Integrate $x^{3/2}$ using the Power Rule.
$$8(2x^{1/2} + C) + \frac{2}{3}x^{3/2} + C + \frac{2}{5}x^{5/2} + C$$
Step 12:
Combine and simplify the terms.
Step 12.1:
Simplify the expression.
$$16x^{1/2} + \frac{2x^{3/2}}{3} + \frac{2x^{5/2}}{5} + C$$
Step 12.2:
Reorder the terms for clarity.
$$16x^{1/2} + \frac{2}{3}x^{3/2} + \frac{2}{5}x^{5/2} + C$$
Step 13:
Present the final antiderivative of $f(x) = \frac{8 + x + x^2}{\sqrt{x}}$.
$$F(x) = 16x^{1/2} + \frac{2}{3}x^{3/2} + \frac{2}{5}x^{5/2} + C$$
Knowledge Notes:
To solve for the antiderivative of a function, we follow these steps:
Integration: The process of finding the antiderivative is called integration.
Indefinite Integral: The general form of an antiderivative is called an indefinite integral, denoted by $\int f(x) \, dx$.
Algebraic Manipulation: Before integrating, we may need to manipulate the function algebraically, such as expressing roots as fractional exponents.
Power Rule for Integration: The Power Rule states that $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ for any real number $n \neq -1$, where $C$ is the constant of integration.
Distributive Property: When integrating a sum, we can integrate each term separately.
Constant Multiple Rule: Constants can be factored out of an integral.
Combining Like Terms: When terms have the same base, we can combine them by adding or subtracting their exponents.
Simplification: After integrating, we simplify the expression and include the constant of integration $C$.
These principles are used to find the antiderivative of a given function by breaking it down into simpler parts that can be integrated individually and then combined to form the final solution.
