Find the Antiderivative f(t)=3t^3+17

Solution:

Step 1:

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

$$F(t)=\int f(t)dt$$

Step 2:

Write down the integral that needs to be solved.

$$F(t)=\int (3t^3+17)dt$$

Step 3:

Decompose the integral into simpler integrals.

$$\int 3t^{3}dt+\int 17dt$$

Step 4:

Extract the constant $3$ from the first integral.

$$3\int t^{3}dt+\int 17dt$$

Step 5:

Apply the power rule for integration to $t^3$.

$$3(\frac{t^4}{4}+C)+\int 17dt$$

Step 6:

Integrate the constant $17$ with respect to $t$.

$$3(\frac{t^4}{4}+C)+17t+C$$

Step 7:

Simplify the expression.

Step 7.1:

Combine the constant $\frac{1}{4}$ with $t^4$.

$$3(\frac{t^4}{4}+C)+17t+C$$

Step 7.2:

Simplify the expression further.

$$\frac{3t^4}{4}+17t+C$$

Step 7.3:

Reorder the terms if necessary.

$$\frac{3t^4}{4}+17t+C$$

Step 8:

Conclude with the antiderivative of the function $f(t)=3t^3+17$.

$$F(t)=\frac{3t^4}{4}+17t+C$$

Knowledge Notes:

To solve for the antiderivative of a polynomial function, one can use the following knowledge points:

1. Indefinite Integral: The antiderivative, or indefinite integral, of a function $f(t)$ is denoted by $\int f(t)dt$ and represents the family of all functions whose derivative is $f(t)$.

2. Linearity of Integration: The integral of a sum of functions is equal to the sum of their integrals. This property allows us to split a complex integral into simpler parts.

3. Constant Multiple Rule: A constant multiple outside the integral can be factored out, which simplifies the integration process.

4. Power Rule for Integration: For any real number $n\ne -1$, the integral of $t^n$ with respect to $t$ is $\frac{t^{n+1}}{n+1}+C$, where $C$ is the constant of integration.

5. Integration of a Constant: The integral of a constant $a$ with respect to $t$ is $at+C$, where $C$ is the constant of integration.

6. Simplification: After integrating, it's important to simplify the expression by combining like terms and constants.

7. Constant of Integration: When finding the indefinite integral, we must add a constant of integration $C$ because the derivative of a constant is zero, and thus the original function could have any constant added to it.