Find the Antiderivative f(x)=(8+x+x^2)/( square root of x)

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:

1. Integration: The process of finding the antiderivative is called integration.

2. Indefinite Integral: The general form of an antiderivative is called an indefinite integral, denoted by $\int f(x)dx$.

3. Algebraic Manipulation: Before integrating, we may need to manipulate the function algebraically, such as expressing roots as fractional exponents.

4. Power Rule for Integration: The Power Rule states that $\int x^{n}dx=\frac{x^{n+1}}{n+1}+C$ for any real number $n\ne -1$, where $C$ is the constant of integration.

5. Distributive Property: When integrating a sum, we can integrate each term separately.

6. Constant Multiple Rule: Constants can be factored out of an integral.

7. Combining Like Terms: When terms have the same base, we can combine them by adding or subtracting their exponents.

8. 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.