Problem 40 - Calculus and its applications

Question

Evaluate. (Be sure to check by differentiating!) $$ \int \frac{3 x^{2} d x}{\left(1+x^{3}\right)^{5}} $$

Step-by-Step Solution

Answer

The integral is \(-\frac{1}{4(1+x^3)^4} + C\). Verification by differentiation confirms the solution.

Step 1: Identify the Substitution

We need to choose a substitution to simplify the integral. Observe the inner function within the denominator. Let's set \( u = 1 + x^3 \). This substitution simplifies the expression and allows us to find a clear derivative.

Step 2: Find the Derivative of the Substitution

Differentiate the substitution expression with respect to \( x \). Thus, \( du = 3x^2 \, dx \). Notice, this matches the numerator of our integral, \( 3x^2 \, dx \), making substitution straightforward.

Step 3: Substitute in the Integral

Convert the integral into terms of \( u \). Substitute \( u = 1 + x^3 \) and \( du = 3x^2 \, dx \) into the integral, changing it to \( \int u^{-5} \, du \).

Step 4: Integrate with Respect to \( u \)

Integrate the expression \( \int u^{-5} \, du \) to get \( \frac{u^{-4}}{-4} + C \). Rewriting gives \( -\frac{1}{4u^4} + C \).

Step 5: Substitute Back to Original Variable

Replace \( u \) with \( 1 + x^3 \) to express the antiderivative in terms of \( x \). This gives the result: \( -\frac{1}{4(1+x^3)^4} + C \).

Step 6: Verify by Differentiating

Differentiate \( -\frac{1}{4(1+x^3)^4} \) with respect to \( x \) to verify. Using the chain rule, this derivative should match the original function \( \frac{3 x^2}{(1+x^3)^5} \). Perform differentiation to confirm correctness.

Key Concepts

Substitution Method

The substitution method is a powerful technique used in calculus to simplify complex integrals. It involves finding a part of the integral, typically a composite function, and replacing it with a single variable. This substitution converts the original integral into a simpler form, allowing for easier evaluation.

Here's how it generally works:

• Identify the inner function within the integral that you can substitute, often a polynomial or composite function.
• Set this inner function equal to a new variable, usually denoted as \( u \). This is your substitution.
• Differentiate the substitution equation with respect to the original variable to find \( du \), the differential of your substitution variable.
• Substitute both the expression for \( u \) and \( du \) into the original integral.

With these steps, a complex integral is transformed into an easier problem that often involves a standard integration formula. Once integrated, it's important to substitute back the original variable, ensuring your answer is in terms of your integral's original format.

Differentiation

Differentiation is the mathematical process used to find the derivative of a function. In integral calculus, differentiation is used to verify whether the antiderivative found through integration is correct.

Here’s how differentiation is typically used:

• Given an antiderivative, differentiate it with respect to the original variable. Use rules like the power rule, the chain rule, and the product rule as needed.
• The result should match the original function inside the integral if your integration was done correctly.

In the context of our integral, once the antiderivative \(-\frac{1}{4(1+x^3)^4} + C\) was obtained, we differentiated it to check for correctness. Using the chain rule, which involves taking the derivative with respect to the inner function \(1 + x^3\), ensures the original integrand is reproduced. This verification step is crucial in confirming that the integration process was correctly executed.

Indefinite Integral

An indefinite integral, or antiderivative, represents a family of functions whose derivative is the given function that appears in the integrand. No limits are specified in indefinite integrals, and they include a constant \( C \) that accounts for all possible constant differences between functions.

• The indefinite integral is denoted by the integral sign \(\int\) followed by the function and the differential symbol. For example, \( \int f(x) \, dx \).
• Finding an indefinite integral involves reversing differentiation, essentially finding what function, when differentiated, would result in the integrand.
• The constant \( C \) is important because differentiation of a constant yields zero, and hence does not appear in the differentiated form.

In our solved example, the indefinite integral \( \int \frac{3x^2}{(1 + x^3)^5} \, dx \) was evaluated using substitution and yielded \( -\frac{1}{4(1+x^3)^4} + C \). Adding \( C \) at the end underscores the possibility of multiple functions differing by a constant that could satisfy the original equation upon differentiation.