Find the Antiderivative f(p)=3(p^2-5)^2(2p)

Solution:

Step 1:

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

$$F(p)=\int f(p)dp$$

Step 2:

Write down the integral that needs to be solved.

$$F(p)=\int 3(p^2-5{)}^2(2p)dp$$

Step 3:

Combine the constants in the integrand.

$$\int 6(p^2-5{)}^{2}pdp$$

Step 4:

Extract the constant multiple from the integral.

$$6\int (p^2-5{)}^{2}pdp$$

Step 5:

Perform a substitution to simplify the integral.

Let $u=p^2-5$. Then calculate $du$.

Step 5.1:

Define the substitution $u=p^2-5$ and find $\frac{du}{dp}$.

Step 5.1.1:

Differentiate $p^2-5$ with respect to $p$.

$$\frac{d}{dp}(p^2-5)$$

Step 5.1.2:

Apply the Sum Rule to differentiate $p^2-5$.

$$\frac{d}{dp}(p^2)+\frac{d}{dp}(-5)$$

Step 5.1.3:

Use the Power Rule, where the derivative of $p^n$ is $n{p}^{n-1}$ for $n=2$.

$$2p+\frac{d}{dp}(-5)$$

Step 5.1.4:

Recognize that the derivative of a constant is zero.

$$2p+0$$

Step 5.1.5:

Combine the terms.

$$2p$$

Step 5.2:

Express the integral in terms of $u$ and $du$.

$$6\int u^2\frac{1}{2}du$$

Step 6:

Combine the squared term with the constant fraction.

$$6\int \frac{u^2}{2}du$$

Step 7:

Factor out the constant from the integral.

$$6(\frac{1}{2}\int u^{2}du)$$

Step 8:

Simplify the expression.

Step 8.1:

Multiply the constants outside the integral.

$$\frac{6}{2}\int u^{2}du$$

Step 8.2:

Reduce the fraction by canceling common factors.

Step 8.2.1:

Factor out the common factor of 2 from 6.

$$\frac{2\cdot 3}{2}\int u^{2}du$$

Step 8.2.2:

Cancel out the common factors.

Step 8.2.2.1:

Factor out the 2 from the denominator.

$$\frac{2\cdot 3}{2\cdot 1}\int u^{2}du$$

Step 8.2.2.2:

Perform the cancellation.

$$\frac{\cancel{2}\cdot 3}{\cancel{2}\cdot 1}\int u^{2}du$$

Step 8.2.2.3:

Rewrite the simplified expression.

$$\frac{3}{1}\int u^{2}du$$

Step 8.2.2.4:

Divide 3 by 1.

$$3\int u^{2}du$$

Step 9:

Apply the Power Rule for integration to find the integral of $u^2$.

$$3(\frac{1}{3}{u}^3+C)$$

Step 10:

Simplify the result.

Step 10.1:

Rewrite the expression.

$$3\left(\frac{1}{3}\right)u^3+C$$

Step 10.2:

Simplify the constant multiple.

Step 10.2.1:

Combine the constants.

$$\frac{3}{3}{u}^3+C$$

Step 10.2.2:

Cancel the common factor of 3.

Step 10.2.2.1:

Perform the cancellation.

$$\frac{\cancel{3}}{\cancel{3}}{u}^3+C$$

Step 10.2.2.2:

Write the simplified expression.

$$u^3+C$$

Step 11:

Substitute back the original variable.

$$(p^2-5{)}^3+C$$

Step 12:

Present the final antiderivative of the function $f(p)=3(p^2-5{)}^2(2p)$.

$$F(p)=(p^2-5{)}^3+C$$

Knowledge Notes:

The process of finding the antiderivative involves several key steps and knowledge points:

1. Indefinite Integral: The antiderivative of a function $f(p)$ is found by integrating the function, denoted as $\int f(p)dp$.

2. Constant Multiple Rule: A constant can be factored out of the integral, simplifying the integration process.

3. Substitution Method: A common technique to simplify integration by introducing a new variable $u$ that represents a part of the original function. This requires calculating the derivative $du$ and expressing the integral in terms of $u$.

4. Sum Rule: The derivative of a sum is the sum of the derivatives.

5. Power Rule for Differentiation: For any real number $n$, the derivative of $p^n$ with respect to $p$ is $n{p}^{n-1}$.

6. Derivative of a Constant: The derivative of a constant is zero.

7. Power Rule for Integration: The integral of $u^n$ with respect to $u$ is $\frac{1}{n+1}{u}^{n+1}$, provided $n\ne -1$.

8. Simplification: Combining like terms, factoring, and canceling common factors are algebraic techniques used to simplify expressions.

9. Back-Substitution: After integrating with respect to the new variable $u$, the original variable is substituted back into the result to obtain the final antiderivative.

10. Integration Constant: When finding an indefinite integral, an arbitrary constant $C$ is added to the result to account for all possible antiderivatives.