Find the Antiderivative f(p)=3(p^2-5)^2(2p)
The given problem is asking to find the antiderivative, also known as the indefinite integral, of a given function. The function is expressed as f(p) = 3(p^2 - 5)^2(2p). The task is to integrate this algebraic expression with respect to the variable p, which means finding a new function F(p) such that the derivative of F(p) with respect to p equals f(p). The process will involve applying integration techniques, which may include the reverse of power rule, product rule, chain rule (for antiderivatives, often called the substitution rule), or a combination of these methods to solve the integral.
$f \left(\right. p \left.\right) = 3 \left(\left(\right. p^{2} - 5 \left.\right)\right)^{2} \left(\right. 2 p \left.\right)$
Answer
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.
$$\int 6(p^2 - 5)^2 p \, dp$$
Step 4:
Extract the constant multiple from the integral.
$$6 \int (p^2 - 5)^2 p \, dp$$
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 $np^{n-1}$ 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.
$$6 \left( \frac{1}{2} \int u^2 du \right)$$
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 $u^2$.
$$3 \left( \frac{1}{3} u^3 + C \right)$$
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:
Indefinite Integral: The antiderivative of a function $f(p)$ is found by integrating the function, denoted as $\int f(p) \, dp$.
Constant Multiple Rule: A constant can be factored out of the integral, simplifying the integration process.
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 $u$.
Sum Rule: The derivative of a sum is the sum of the derivatives.
Power Rule for Differentiation: For any real number $n$, the derivative of $p^n$ with respect to $p$ is $np^{n-1}$.
Derivative of a Constant: The derivative of a constant is zero.
Power Rule for Integration: The integral of $u^n$ with respect to $u$ is $\frac{1}{n+1}u^{n+1}$, provided $n \neq -1$.
Simplification: Combining like terms, factoring, and canceling common factors are algebraic techniques used to simplify expressions.
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.
Integration Constant: When finding an indefinite integral, an arbitrary constant $C$ is added to the result to account for all possible antiderivatives.
