Problem 6 - Applied Calculus

Question

Find the integrals .Check your answers by differentiation. $$\int 2 x\left(x^{2}+1\right)^{5} d x$$

Step-by-Step Solution

Answer

Integral is \( \frac{(x^2+1)^6}{6} + C \). Differentiation confirms it.

Step 1: Identify the Integration Method

The integral you need to solve is \( \int 2x (x^2 + 1)^5 \ dx \). To solve this, we can use substitution because the derivative of \(x^2 + 1\) is \(2x\), which is present in the integrand.

Step 2: Make a Substitution

Let \( u = x^2 + 1 \). Then, differentiate \( u \) with respect to \( x \) to get \( du/dx = 2x \) or \( du = 2x \ dx \). This substitution transforms the integral into \( \int u^5 \, du \).

Step 3: Integrate in Terms of u

Now integrate \( \int u^5 \, du \). This is a basic power function integral, which results in \( \frac{u^6}{6} + C \) where \( C \) is the constant of integration.

Step 4: Substitute Back in Terms of x

Replace \( u \) with \( x^2 + 1 \). Thus, the integral becomes \( \frac{(x^2 + 1)^6}{6} + C \).

Step 5: Differentiate the Result to Verify

To check your answer by differentiation, differentiate \( \frac{(x^2 + 1)^6}{6} + C \) with respect to \( x \). Using the chain rule, the derivative is \( 6(x^2 + 1)^5 \cdot 2x / 6 \), which simplifies to \( 2x(x^2 + 1)^5 \). This matches the original integrand.

Key Concepts

Understanding the Substitution Method

The substitution method is a powerful tool in integral calculus, particularly when dealing with integrals that seem complex or are challenging to solve directly. The main idea is to simplify the integral by introducing a new variable that converts the integral into a more manageable form. This technique leverages the relationship between the derivative and the original function inside the integral.
\( \int 2x (x^2 + 1)^5 \ dx \) is a perfect candidate for substitution because the derivative of \(x^2 + 1\) is exactly \(2x\), which is part of the integrand. For substitution, the steps are as follows:

• Choose a substitution, in this case, let \( u = x^2 + 1 \).
• Find \( du \) by differentiating \( u \) with respect to \( x \), resulting in \( du = 2x \ dx \).
• Replace all \( x \)-terms in the integral with \( u \)-terms, transforming the integral into \( \int u^5 \, du \).

This transformation simplifies the integration process significantly, converting what seemed like a difficult integral into a straightforward polynomial one.

Integrating and Verification of Result

Integration verification is a crucial part of integral calculus as it ensures the correctness of the solution. After changing variables with substitution, integrate the new expression to find the antiderivative. Integration sums up to finding a function whose derivative matches the original function inside the integrand.
Upon performing the integration of \( \int u^5 \, du \), we obtain \( \frac{u^6}{6} + C \). We then substitute back the original variable:\( u = x^2 + 1 \), leading to the solution \( \frac{(x^2 + 1)^6}{6} + C \).
To verify the accuracy, differentiate the result \( \frac{(x^2 + 1)^6}{6} + C \) with respect to \( x \). Correct differentiation should yield the initial integrand \( 2x(x^2 + 1)^5 \), indicating the solution was computed accurately. This verification step is like doing a consistency check for your work, ensuring every step logically leads to the expected mathematical outcome.

Chain Rule and Differentiation

Differentiation using the chain rule involves finding the derivative of a composite function, like \( (x^2 + 1)^6 \). In calculus, the chain rule is vital for differentiating composed functions, and it's essential when verifying integration results.
For our problem, once the integral is solved, the chain rule is utilized to differentiate \( \frac{(x^2 + 1)^6}{6} + C \) back into the original function. When differentiating \( (x^2 + 1)^6 \), the chain rule requires multiplying 6 by the derivative of the inner function \( x^2 + 1 \), which is \( 2x \). This is expressed in this calculation: \( 6(x^2 + 1)^5 \cdot 2x / 6 \).
This process confirms that upon integrating and substituting, the solution matches the problem's initial condition. Understanding this concept ensures that differentiation and integration are inverses of each other, forming a foundational aspect of integral calculus methods like substitution.