Find the Antiderivative g(x)=1/(2 square root of x)+(x^2)/4+C

Solution:

Step 1:

Identify the antiderivative $G(x)$ by integrating the given function $g(x)$.

$$G(x) = \int g(x) \, dx$$

Step 2:

Write out the integral that needs to be solved.

$$G(x) = \int \left( \frac{1}{2\sqrt{x}} + \frac{x^2}{4} + C \right) dx$$

Step 3:

Decompose the integral into separate integrals for each term.

$$\int \frac{1}{2\sqrt{x}} \, dx + \int \frac{x^2}{4} \, dx + \int C \, dx$$

Step 4:

Extract the constant $\frac{1}{2}$ from the integral as it is not dependent on $x$.

$$\frac{1}{2} \int \frac{1}{\sqrt{x}} \, dx + \int \frac{x^2}{4} \, dx + \int C \, dx$$

Step 5:

Apply the rules for exponents.

Step 5.1:

Express $\sqrt{x}$ as $x^{\frac{1}{2}}$.

$$\frac{1}{2} \int x^{-\frac{1}{2}} \, dx + \int \frac{x^2}{4} \, dx + \int C \, dx$$

Step 5.2:

Rewrite the integral with the exponent in the numerator.

$$\frac{1}{2} \int x^{-\frac{1}{2}} \, dx + \int \frac{x^2}{4} \, dx + \int C \, dx$$

Step 5.3:

Simplify the expression by combining the exponents.

Step 5.3.1:

Apply the power rule $(a^m)^n = a^{mn}$.

$$\frac{1}{2} \int x^{-\frac{1}{2}} \, dx + \int \frac{x^2}{4} \, dx + \int C \, dx$$

Step 5.3.2:

Combine the constants.

$$\frac{1}{2} \int x^{-\frac{1}{2}} \, dx + \int \frac{x^2}{4} \, dx + \int C \, dx$$

Step 5.3.3:

Adjust the negative exponent.

$$\frac{1}{2} \int x^{-\frac{1}{2}} \, dx + \int \frac{x^2}{4} \, dx + \int C \, dx$$

Step 6:

Integrate $x^{-\frac{1}{2}}$ using the Power Rule.

$$\frac{1}{2} (2x^{\frac{1}{2}} + C) + \int \frac{x^2}{4} \, dx + \int C \, dx$$

Step 7:

Extract the constant $\frac{1}{4}$ from the integral.

$$\frac{1}{2} (2x^{\frac{1}{2}} + C) + \frac{1}{4} \int x^2 \, dx + \int C \, dx$$

Step 8:

Integrate $x^2$ using the Power Rule.

$$\frac{1}{2} (2x^{\frac{1}{2}} + C) + \frac{1}{4} \left(\frac{x^3}{3} + C\right) + \int C \, dx$$

Step 9:

Apply the constant rule to integrate $C$.

$$\frac{1}{2} (2x^{\frac{1}{2}} + C) + \frac{1}{4} \left(\frac{x^3}{3} + C\right) + Cx + C$$

Step 10:

Simplify the expression.

Step 10.1:

Simplify the terms.

$$x^{\frac{1}{2}} + \frac{x^3}{12} + Cx + C$$

Step 10.2:

Reorder the terms for the final antiderivative.

$$x^{\frac{1}{2}} + \frac{1}{12} x^3 + Cx + C$$

Step 11:

The antiderivative $G(x)$ is the result of integrating the function $g(x)$.

$$G(x) = x^{\frac{1}{2}} + \frac{1}{12} x^3 + Cx + C$$

Knowledge Notes:

To solve for the antiderivative of a given function, we follow these steps:

1. Integration: The process of finding the antiderivative is known as integration. The antiderivative is a function whose derivative is the original function.

2. Indefinite Integral: The antiderivative is also referred to as the indefinite integral and is represented by the integral sign followed by the function and the differential, e.g., $\int f(x) \, dx$.

3. Linearity of Integration: The integral of a sum of functions is equal to the sum of their integrals. This allows us to split the integral of multiple terms into separate integrals.

4. Constants: Constants can be factored out of the integral since they do not depend on the variable of integration.

5. Exponent Rules: Exponent rules are used to rewrite expressions in a form that is easier to integrate, such as converting a square root to a fractional exponent.

6. Power Rule for Integration: The Power Rule states that the integral of $x^n$ with respect to $x$ is $\frac{x^{n+1}}{n+1}$, provided $n \neq -1$.

7. Constant of Integration: When finding the indefinite integral, we add a constant of integration, $C$, because the derivative of a constant is zero, and there are infinitely many antiderivatives differing by a constant.

8. Simplification: After integrating, we simplify the expression to obtain the final antiderivative. This may include combining like terms and rearranging the terms into a standard form.