Problem 39 - Calculus and its applications

Question

Evaluate. (Be sure to check by differentiating!) $$ \int \frac{x^{3} d x}{\left(2-x^{4}\right)^{7}} $$

Step-by-Step Solution

Answer

The evaluated integral is \( \frac{1}{24} (2 - x^4)^{-6} + C \).

Step 1: Identify the Integral Type

The given integral is \( \int \frac{x^3}{(2-x^4)^7} \, dx \). It is a type of rational function where substitution might simplify the problem. Notice that the derivative of \( x^4 \) resembles the numerator.

Step 2: Choose a Suitable Substitution

Let \( u = 2 - x^4 \). Then, differentiating both sides with respect to \( x \) gives \( du = -4x^3 \, dx \), which implies \( x^3 \, dx = -\frac{1}{4} du \).

Step 3: Rewrite the Integral with Substitution

Substituting \( u \) and \( du \) into the integral, we have:\[ \int \frac{x^3}{(2-x^4)^7} \, dx = \int \frac{-1/4 \, du}{u^7} \]. This simplifies to:\[ -\frac{1}{4} \int u^{-7} \, du \].

Step 4: Integrate using Power Rule

Using the power rule for integration, which states \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \) for \( n eq -1 \), apply to \( u^{-7} \):\[ -\frac{1}{4} \cdot \frac{u^{-6}}{-6} = \frac{1}{24} u^{-6} + C \].

Step 5: Substitute Back for \( u \)

Substitute back \( u = 2 - x^4 \) into the antiderivative:\[ \frac{1}{24} (2 - x^4)^{-6} + C \].

Step 6: Differentiate to Confirm

Differentiate \( \frac{1}{24} (2 - x^4)^{-6} + C \) with respect to \( x \). Using the chain rule: 1. \( -6(2-x^4)^{-7}(-4x^3) \times \frac{1}{24} = \frac{x^3}{(2-x^4)^7} \).This confirms the original integrand, verifying the integration is correct.

Key Concepts

Substitution Method

The substitution method is an essential technique in integral calculus aimed at simplifying the process of finding antiderivatives. This method is particularly effective when dealing with complex integrals by transforming them into a simpler form. In our example, the original integral is\[ \int \frac{x^3}{(2-x^4)^7} \, dx \]To apply the substitution method, we choose a part of the integrand to substitute with a new variable, often called "\(u\)". Here, we set \( u = 2 - x^4 \). This choice makes the integration process more manageable because the derivative of \(x^4\) (i.e., \(-4x^3\)) closely resembles the \(x^3\) term in the numerator.The next step is differentiating \(u\) to find \(du\). This gives \( du = -4x^3 \, dx \). Rearranging yields \( x^3 \, dx = -\frac{1}{4} du \). By substituting these into the integral, the complex expression is reduced to a simpler form:\[ \int \frac{-1/4 \, du}{u^7} \]Using the substitution method turns the original problem, into a straightforward power integral. This technique is invaluable for tackling integrals with integrands that neatly fit the form of the function and its derivative.

Power Rule for Integration

The power rule for integration is one of the fundamental techniques of calculus, useful for finding antiderivatives of power functions. It states that for any real number \(n\) other than \(-1\), the integral of \(u^n\) with respect to \(u\) is:\[ \int u^n \, du = \frac{u^{n+1}}{n+1} + C \]where \(C\) is the constant of integration. In our specific exercise, after applying the substitution method, we had the integral\[ \int u^{-7} \, du \]Using the power rule, we integrate \( u^{-7} \) to:\[ \frac{u^{-6}}{-6} + C \]Multiplying by the constant factor obtained from substitution, \(-\frac{1}{4}\), results in:\[ \frac{1}{24} u^{-6} + C \]This solution showcases the direct application of the power rule to the new simplified integral, allowing us to solve the problem efficiently. The power rule is a simple yet powerful tool in integral calculus and is specifically helpful when handling integrals of polynomial expressions.

Chain Rule in Differentiation

The chain rule is a pivotal concept in differentiation, particularly useful when differentiating compositions of functions. After integrating, verifying your work via differentiation checks your solutions' correctness. In our problem, to confirm that the integration was done correctly, differentiation of the result is required:\[ \frac{1}{24} (2 - x^4)^{-6} + C \]Using the chain rule in differentiation, start with the outer function, \((2 - x^4)^{-6}\). Differentiate it with respect to \(2 - x^4\), giving \(-6(2-x^4)^{-7}\). Now, multiply this by the derivative of the inner function \(2 - x^4\), which is \(-4x^3\). Combine these:\[ -6(2-x^4)^{-7}(-4x^3) \]This simplifies to:\[ 24x^3(2-x^4)^{-7} \]Finally, multiplying by the outside constant factor, \(\frac{1}{24}\), returns to our original integrand,\[ \frac{x^3}{(2-x^4)^7} \]This verifies that the integration was performed accurately. The chain rule is indispensable, ensuring our integration results are robust and correct.