Question

On which derivative rule is integration by parts based?

Short answer

Answer: Integration by Parts is based on the Product Rule for differentiation. It allows us to integrate products of functions by breaking them down into simpler integrals, using the formula: $$\int u^{\prime }(x)v(x)dx=u(x)v(x)-\int u(x)v^{\prime }(x)dx$$.

Step-by-step solution

Identify the derivative rule

Integration by parts is based on the Product Rule for differentiation, which states that the derivative of a product of two functions u(x) and v(x) is given by: $$(u(x)v(x){)}^{\prime }=u^{\prime }(x)v(x)+u(x)v^{\prime }(x)$$

Reverse the Product Rule

To reverse the Product Rule to an integration formula, we'll start by considering the derivative: $$\frac{d}{dx}(u(x)v(x))=u^{\prime }(x)v(x)+u(x)v^{\prime }(x)$$ Now, we'll integrate both sides of the equation with respect to x: $$\int \frac{d}{dx}(u(x)v(x))dx=\int (u^{\prime }(x)v(x)+u(x)v^{\prime }(x))dx$$ The left side integrates the derivative, leaving us with the original product of functions: $$u(x)v(x)=\int (u^{\prime }(x)v(x)+u(x)v^{\prime }(x))dx$$

Apply the Integration by Parts Formula

Now we can rearrange the equation to solve for one of the integrals: $$\int u^{\prime }(x)v(x)dx=u(x)v(x)-\int u(x)v^{\prime }(x)dx$$ And that's the Integration by Parts formula! This is based on the Product Rule for differentiation, and allows us to integrate products of functions by breaking them down into simpler integrals.

Key concepts

Product Rule

The Product Rule is a fundamental concept in calculus used when differentiating a product of two functions. Suppose you have two functions, say $u(x)$ and $v(x)$. The Product Rule describes how to find the derivative of their product. Instead of differentiating each function separately, the rule tells us to take the derivative of the first function, multiply it by the second function, and then add it to the first function multiplied by the derivative of the second. In symbolic form, this is expressed as:- $(u(x)v(x){)}^{\prime }=u^{\prime }(x)v(x)+u(x)v^{\prime }(x)$.Understanding this rule is crucial for breaking down complex expressions and is the basis for more advanced techniques like integration by parts.Remember:- Differentiate the first function and multiply it by the second.- Add it to the first function multiplied by the derivative of the second.

Differentiation

Differentiation is a calculus technique used to find the rate at which a function is changing at any given point. Think of it as a tool for measuring how one quantity changes as another quantity changes, often described as finding the slope of a curve at a point. a great example is determining the speed of a car at a particular moment, given its distance over time. When you differentiate a function, you apply certain rules (like the Product Rule) to calculate its derivative, which gives you the function's rate of change. Key tips: - Differentiation often involves rules such as power, chain, and product rules. - It transforms a function into its derivative, facilitating analysis and problem-solving in calculus.

Calculus

Calculus is the branch of mathematics focusing on change and motion. It is divided mainly into two parts: differentiation and integration. Each part helps us understand different aspects of changing systems. Differentiation, as discussed earlier, looks at rates of change, while integration deals with calculating areas under curves or the accumulation of quantities. In real-world problems, calculus helps in areas such as: - physics for motion and forces, - engineering for stress and material design, - economics for maximizing functions like profits and minimizing costs. This field requires understanding limits, functions, and continuity, forming the backbone of understanding dynamic systems.

Integration

Integration is an essential technique in calculus that can be thought of as the reverse process of differentiation. While differentiation breaks down functions into rates of change, integration helps to combine these rates to calculate the total accumulated quantity. This could include areas under curves, volumes, or even more abstract quantities like probabilities.Integration by parts specifically applies the reasoning behind the Product Rule to simplify integrating products of functions. This method involves rearranging the differentiation rule and helps in solving integrals that are otherwise challenging.Remember:- Integration by Parts formula: $\int u^{\prime }(x)v(x)dx=u(x)v(x)-\int u(x)v^{\prime }(x)dx$.- It can dissect products into manageable pieces for simpler and efficient calculations.