Find the Antiderivative f(x)=x^3+x

Solution:

Step 1:

Identify the antiderivative $F(x)$ by integrating the given function $f(x)$.

$$F(x)=\int f(x)dx$$

Step 2:

Write down the integral that needs to be solved.

$$F(x)=\int (x^3+x)dx$$

Step 3:

Decompose the integral into the sum of two separate integrals.

$$\int x^{3}dx+\int xdx$$

Step 4:

Apply the Power Rule to integrate $x^3$ with respect to $x$, which results in $\frac{x^4}{4}$.

$$\frac{x^4}{4}+C_1+\int xdx$$

Step 5:

Again, apply the Power Rule to integrate $x$ with respect to $x$, which gives $\frac{x^2}{2}$.

$$\frac{x^4}{4}+C_1+\frac{x^2}{2}+C_2$$

Step 6:

Combine the terms and constants.

$$\frac{x^4}{4}+\frac{x^2}{2}+C$$

Step 7:

Present the final antiderivative of $f(x)=x^3+x$.

$$F(x)=\frac{x^4}{4}+\frac{x^2}{2}+C$$

Knowledge Notes:

The process of finding the antiderivative, also known as the indefinite integral, involves reversing the process of differentiation. The antiderivative of a function $f(x)$ is another function $F(x)$ such that $F^{\prime }(x)=f(x)$. The constant $C$ represents the constant of integration, which is added because the derivative of a constant is zero.

The Power Rule for integration states that for any real number $n\ne -1$, the integral of $x^n$ with respect to $x$ is given by:

$$\int x^{n}dx=\frac{x^{n+1}}{n+1}+C$$

When integrating a sum of functions, the integral can be split into the sum of integrals of each term:

$$\int [f(x)+g(x)]dx=\int f(x)dx+\int g(x)dx$$

In this problem, we used the Power Rule to find the antiderivatives of $x^3$ and $x$, and then combined the results, including the constant of integration, to obtain the final antiderivative of the given function.