Problem 29 - Applied Calculus

Question

Find each indefinite integral by the substitution method or state that it cannot be found by our substitution formulas. $$ \int\left(2 y^{2}+4 y\right)^{5}(y+1) d y $$

Step-by-Step Solution

Answer

The indefinite integral is \( \frac{1}{24} (2y^2 + 4y)^6 + C \).

Step 1: Identify the Substitution

For the given integral \( \int (2y^2 + 4y)^5 (y + 1) \, dy \), we notice the inside of the parentheses \((2y^2 + 4y)^5\) can be a good candidate for substitution. Let's set \( u = 2y^2 + 4y \).

Step 2: Calculate the Derivative

Differentiate \( u \,= 2y^2 + 4y \) with respect to \( y \). This gives \( \frac{du}{dy} = 4y + 4 \).

Step 3: Express \( dy \) in Terms of \( du \)

Solve for \( dy \) in terms of \( du \): \( du = (4y + 4) \, dy \) or \( dy = \frac{du}{4(y + 1)} \).

Step 4: Substitute in the Integral

Substitute \( u \) and \( dy \) into the original integral. The integral becomes \( \int u^5 \cdot (y + 1) \cdot \frac{du}{4(y + 1)} \). Simplify to \( \int \frac{1}{4} u^5 \, du \).

Step 5: Simplify and Integrate

The integral \( \int \frac{1}{4} u^5 \, du \) simplifies to \( \frac{1}{4} \cdot \frac{u^6}{6} + C = \frac{1}{24} u^6 + C \).

Step 6: Back-Substitute for \( y \)

Replace \( u \) with the expression in terms of \( y \): \( u = 2y^2 + 4y \). The final result is \( \frac{1}{24} (2y^2 + 4y)^6 + C \).

Key Concepts

Substitution Method

The substitution method is a widely used technique in calculus, especially for finding indefinite integrals. This technique simplifies complex integrals by substituting a part of the integral with a single variable, usually denoted as 'u'.

Here's a simple process to follow:

• Choose a part of the integral to set equal to 'u'. This is often the inside of a complicated expression.
• Differentiate 'u' with respect to your variable, say 'y', to find \( \frac{du}{dy} \).
• Rewrite 'dy' in terms of 'du' and substitute all occurrences of 'y' in the integral with expressions involving 'u'.
• Simplify the integral, and then integrate with respect to 'u'.
• Finally, back-substitute the original 'y' expressions to return to the variable 'y'.

In our example, we used substitution to simplify \( (2y^2 + 4y)^5 \) into a single variable \( u \), making the problem much more manageable.

Calculus

Calculus is the branch of mathematics that deals with rates of change and motion. It connects algebraic and geometric concepts to explore how mathematical systems change over time or space. There are two main branches:

• Differential Calculus, which involves the computation of derivatives to find tangent lines or instantaneous rates of change.
• Integral Calculus, which concerns itself with determining the area under curves, volumes of solids, and quantities accumulated over time.

In integral calculus, the primary aim is often to find the antiderivative or indefinite integral, such as in our given exercise. This process effectively reverses differentiation.

By using calculus, specifically integration, and the substitution method, we were able to find the indefinite integral \( \int (2y^2+4y)^5(y+1) \, dy \). Calculus provides a framework for solving such problems and understanding the behavior of mathematical functions.

Integration Techniques

Integration techniques are methods used to evaluate integrals, which are essential tools in calculus. The choice of technique often depends on the form of the function you are integrating. Here are a few common techniques:

• Substitution Method: This is useful when the integral contains a function and its derivative. This is the technique we used in our problem.
• Integration by Parts: Best for integrals of products of functions, based on the product rule for differentiation.
• Partial Fractions Decomposition: Useful for rational functions where the degree of the numerator is less than the degree of the denominator.
• Trigonometric Substitution: Effective for integrals involving the square roots of quadratic expressions.

Practicing different techniques helps to solve increasingly complex integrals. In this exercise, the substitution method was the most fitting. By identifying parts of the integral that can be defined as 'u', it became more straightforward to find the antiderivative of the given expression.