Integration by Substitution: Antiderivative of ( e^{p^3} p^2 )

Introduction to Integration by Substitution:

In calculus, integration by substitution is a technique for finding antiderivatives and integrals. By making an appropriate substitution, we can simplify the integral and make the problem more manageable. This article will demonstrate how to find the antiderivative of the function ( e^{p^3} p^2 ) using the substitution method. We will also verify the solution using the chain rule and the fundamental theorem of calculus.

Step-by-Step Integration by Substitution

Let's start with the integral:

1. Choose the Substitution: Let ( u p^3 ). This is a strategic choice because the derivative of ( p^3 ) appears in the original integral as ( 3p^2 ).

2. Find the Differential: Differentiate both sides with respect to ( p ).

$u p^3$ implies that $du 3p^2 dp$.

3. Isolate the Differential: To match the integral exactly, we need to solve for ( p^2 dp ). Divide both sides of the equation by 3.

$frac{1}{3} du p^2 dp$.

4. Substitute and Integrate: Rewrite the integral in terms of ( u ) and integrate.

$int e^{p^3} p^2 dp int e^u frac{1}{3} du frac{1}{3} int e^u du$.

5. Integrate the Exponential Function: The integral of ( e^u ) with respect to ( u ) is ( e^u ), so:

$frac{1}{3} int e^u du frac{1}{3} e^u C$, where ( C ) is the constant of integration.

6. Substitute Back: Replace ( u ) with ( p^3 ).

$frac{1}{3} e^u C frac{1}{3} e^{p^3} C$.

Verification Using the Chain Rule

To verify that the antiderivative is correct, we can take the derivative of the result:

Derivative of ( frac{1}{3} e^{p^3} C ) with respect to ( p ):

$frac{d}{dp} left( frac{1}{3} e^{p^3} right)$.

Using the chain rule, where the derivative of ( e^{p^3} ) is ( e^{p^3} cdot 3p^2 ), we get:

$frac{1}{3} cdot 3e^{p^3} p^2 e^{p^3} p^2$.

This confirms that the antiderivative is indeed ( frac{1}{3} e^{p^3} C ).

Conclusion

Integration by substitution is a powerful tool that simplifies complex integrals. By choosing an appropriate substitution and applying the chain rule, we can find the antiderivative of ( e^{p^3} p^2 ) as ( frac{1}{3} e^{p^3} C ).

Understanding integration by substitution, the antiderivative, and the chain rule is crucial for solving a wide range of calculus problems.

If you have any further questions or need more examples, feel free to ask!