Problem 17 - Applied Calculus

Question

Find the integrals .Check your answers by differentiation. $$\int x\left(x^{2}+3\right)^{2} d x$$

Step-by-Step Solution

Answer

The integral is \( \frac{1}{6}(x^2 + 3)^3 + C \). It is verified by differentiation.

Step 1: Identify the Method of Integration

We have the integral \( \int x(x^2 + 3)^2 \, dx \). The presence of \( x \) and \( (x^2 + 3)^2 \) suggests using substitution. Let \( u = x^2 + 3 \), which simplifies the expression.

Step 2: Calculate the Derivative for Substitution

Differentiate \( u = x^2 + 3 \) to find \( du \). We get \( du = 2x \, dx \), hence \( x \, dx = \frac{1}{2}du \). This allows the integral to be rewritten in terms of \( u \) instead of \( x \).

Step 3: Substitute and Simplify the Integral

Substituting \( u = x^2 + 3 \) and \( x \, dx = \frac{1}{2}du \) into the integral, we have:\[\int x(x^2 + 3)^2 \, dx = \int (x \cdot u^2) \, dx = \frac{1}{2} \int u^2 \, du.\] This integral is now simpler to evaluate.

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

Integrate \( \frac{1}{2} \int u^2 \, du \) to get the antiderivative. The integral of \( u^2 \) is \( \frac{u^3}{3} \), so we have:\[\frac{1}{2} \left( \frac{u^3}{3} \right) = \frac{1}{6}u^3 + C\]where \( C \) is the constant of integration.

Step 5: Substitute Back in Terms of \( x \)

Replace \( u \) back with \( x^2 + 3 \) to express the antiderivative in terms of \( x \). This yields:\[\frac{1}{6}(x^2 + 3)^3 + C.\] This is the required indefinite integral.

Step 6: Verify the Solution by Differentiation

Differentiate \( \frac{1}{6}(x^2 + 3)^3 + C \) with respect to \( x \) to check our integration:- First, use the chain rule: \( \frac{d}{dx}[(x^2+3)^3] = 3(x^2+3)^2 \times 2x = 6x(x^2+3)^2 \).- The derivative of \( \frac{1}{6}(x^2+3)^3 + C \) is \( x(x^2 + 3)^2 \), which matches the original integrand.

Key Concepts

Substitution Method

The substitution method is a powerful technique used in calculus to evaluate more complex integrals. This process simplifies the expression by replacing one part of it with a single-variable substitute. In our exercise, we started with the integral \( \int x(x^2 + 3)^2 \, dx \). Observing that \( (x^2 + 3)^2 \) can be simplified, we introduced a substitution variable \( u \), setting \( u = x^2 + 3 \). This transformation is essential because it allows us to change the variable of integration to \( u \).

• This simplification reduces the complexity of the integral.
• It transforms the original integral into a polynomial in terms of \( u \), which is easier to manage.

In general, the substitution method is handy when you recognize a function and its derivative within the integral.

Indefinite Integral

An indefinite integral represents a family of functions and is the reverse process of differentiation. Unlike definite integrals, indefinite integrals do not have specified bounds. They yield a general antiderivative plus a constant of integration, \( C \).For our problem, we aimed to find the indefinite integral \( \int x(x^2 + 3)^2 \, dx \), which is a function that differentiates back to the original expression. By employing the substitution method, we transformed and solved the integral, eventually obtaining:\[ \frac{1}{6}(x^2 + 3)^3 + C. \]

• Indefinite integrals are essential in capturing all possible antiderivatives of the expression.
• They play a crucial role in solving differential equations and solving problems involving motion and areas under curves.

Always keep the constant of integration to account for any shifts in the original function.

Antiderivative

An antiderivative of a function is a function whose derivative yields the original function. The process of finding an antiderivative is termed integration.In the given exercise, finding the antiderivative involved rewriting \( \frac{1}{2} \int u^2 \, du \) and integrating it.

• By calculating the antiderivative of \( u^2 \), we get \( \frac{u^3}{3} \).
• Multiplying by the outside constant results in \( \frac{1}{6}u^3 + C \).

Here's the vital part: always replace back any substitutions made initially. So, \( u \) turns back into \( x^2+3 \), giving us the specific antiderivative in terms of \( x \).Seeking an antiderivative is akin to asking what '{some} function' must be so that differentiating it returns the integrand in question.

Chain Rule

The chain rule is a fundamental principle in calculus, used when differentiating composite functions. It states that the derivative of \( f(g(x)) \) is \( f'(g(x)) \cdot g'(x) \).In our exercise, the chain rule was crucial for verification. By differentiating the antiderivative \( \frac{1}{6}(x^2+3)^3 + C \) using the chain rule, we should return to our original integrand: \( x(x^2 + 3)^2 \).

• The differentiation of \( (x^2+3)^3 \) requires the chain rule.
• Identify outer and inner functions: here, the outer is \( t^3 \) and the inner is \( t = x^2 + 3 \).
• The derivative is \( 3(x^2+3)^2 \cdot 2x = 6x(x^2+3)^2 \).

Using the chain rule confirms our antiderivative was correct, showing the robustness and reliability of this computation method.