Find the Antiderivative g(u)=2u+ fourth root of u

Solution:

Step 1:

Identify the antiderivative $G(u)$ by integrating the function $g(u)$.
$G(u) = \int g(u) \, du$

Step 2:

Write down the integral that needs to be solved.
$G(u) = \int (2u + u^{\frac{1}{4}}) \, du$

Step 3:

Decompose the integral into two separate integrals.
$\int 2u \, du + \int u^{\frac{1}{4}} \, du$

Step 4:

Extract the constant $2$ from the first integral.
$2 \int u \, du + \int u^{\frac{1}{4}} \, du$

Step 5:

Apply the Power Rule to integrate $u$.
$2 \left( \frac{u^2}{2} + C \right) + \int u^{\frac{1}{4}} \, du$

Step 6:

Express the fourth root of $u$ as $u$ raised to the power of $\frac{1}{4}$.
$2 \left( \frac{u^2}{2} + C \right) + \int u^{\frac{1}{4}} \, du$

Step 7:

Integrate $u^{\frac{1}{4}}$ using the Power Rule.
$2 \left( \frac{u^2}{2} + C \right) + \frac{4}{5} u^{\frac{5}{4}} + C$

Step 8:

Simplify the expression.

Step 8.1:

Combine the terms involving $u^2$.
$2 \left( \frac{u^2}{2} + C \right) + \frac{4}{5} u^{\frac{5}{4}} + C$

Step 8.2:

Final simplification.
$u^2 + \frac{4}{5} u^{\frac{5}{4}} + C$

Step 9:

Conclude with the antiderivative of $g(u) = 2u + u^{\frac{1}{4}}$.
$G(u) = u^2 + \frac{4}{5} u^{\frac{5}{4}} + C$

Knowledge Notes:

The process of finding the antiderivative, also known as the indefinite integral, involves reversing the differentiation process. Here are the relevant knowledge points used in the solution:

1. Indefinite Integral: The antiderivative or indefinite integral of a function $f(x)$ is a function $F(x)$ whose derivative is $f(x)$. It is represented as $\int f(x) \, dx = F(x) + C$, where $C$ is the constant of integration.

2. Power Rule for Integration: If $n \neq -1$, then $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$. This rule is used to integrate polynomials and other functions with exponents.

3. Constant Multiple Rule: A constant factor can be pulled out of the integral, i.e., $\int k \cdot f(x) \, dx = k \int f(x) \, dx$, where $k$ is a constant.

4. Simplification: After integrating, terms are combined and simplified to express the antiderivative in its simplest form.

5. Notation: The notation $\int$ denotes the integral sign, and $du$ or $dx$ indicates the variable of integration.

6. Roots as Exponents: Roots can be rewritten as fractional exponents, for example, $\sqrt[n]{x} = x^{\frac{1}{n}}$.

By applying these principles, the antiderivative of the given function $g(u) = 2u + \sqrt[4]{u}$ can be found step by step.