Question

Verify the following indefinite integrals by differentiation. These integrals are derived in later chapters. $$\int \frac{x}{\sqrt{x^{2}+1}} d x=\sqrt{x^{2}+1}+C$$

Short answer

Question: Verify that the given indefinite integral is correct: $$\int \frac{x}{\sqrt{x^{2}+1}} d x=\sqrt{x^{2}+1}+C$$ Answer: After differentiating the given function, we obtained its derivative to be $$f'(x) = \frac{x}{\sqrt{x^2 + 1}}$$, which matches the original integrand. Thus, the indefinite integral is verified to be correct.

Step-by-step solution

Identify the function to differentiate

We are given the indefinite integral: $$\int \frac{x}{\sqrt{x^{2}+1}} d x=\sqrt{x^{2}+1}+C$$ The function to differentiate is: $$f(x) = \sqrt{x^2 + 1} + C$$

Differentiate the function

To differentiate the function, we need to apply the chain rule. The chain rule states that if we have a composite function such as $$h(x) = f(g(x))$$, then $$h'(x) = f'(g(x)) \cdot g'(x)$$ Notice that our function f(x) can be rewritten as: $$f(x) = (\sqrt{u})$$ with $$u = x^2 + 1$$ Differentiate both parts: 1. Differentiate the outer function (\(\sqrt{u}\)): $$\frac{d}{du}(\sqrt{u}) = \frac{1}{2\sqrt{u}}$$ 2. Differentiate the inner function (\(x^2 + 1\)): $$\frac{d}{dx}(x^2 + 1) = 2x$$ Now, apply the chain rule: $$f'(x) = \frac{1}{2 \sqrt{x^2 + 1}} \cdot 2x$$

Simplify the result

Now, we simplify the derivative: $$f'(x) = \frac{2x}{2\sqrt{x^2 + 1}} = \frac{x}{\sqrt{x^2 + 1}}$$

Compare the derivative to the original integrand

The derivative we obtained is: $$f'(x) = \frac{x}{\sqrt{x^2 + 1}}$$ This matches the original integrand, so the indefinite integral is verified: $$\int \frac{x}{\sqrt{x^{2}+1}} d x=\sqrt{x^{2}+1}+C$$

Key concepts

Chain Rule Differentiation

Understanding the chain rule in differentiation is crucial for dealing with composite functions, where one function is nested within another. The chain rule helps us to differentiate a function of a function, such as \(h(x) = f(g(x))\). The essence of the chain rule is that if you want to differentiate \(h(x)\) with respect to \(x\), you first differentiate the outer function \(f\) with respect to the inner function \(g\), then multiply that by the derivative of the inner function \(g\) with respect to \(x\). In formula terms, this is \[h'(x) = f'(g(x)) \cdot g'(x)\].

Think of it like peeling an onion. You first peel the outer layer (differentiate the outer function) and then peel the next layer underneath (differentiate the inner function). It's a step-by-step procedure that requires you to be orderly and methodical. For the problem \(\int \frac{x}{\sqrt{x^{2}+1}} dx\), we used the chain rule to differentiate the result of the integral, \(\sqrt{x^{2}+1} + C\), confirming that differentiation of the solution matches the integrand.

Integrals Calculus

In integrals calculus, indefinite integrals are a fundamental concept. An indefinite integral represents a family of functions that, when differentiated, would yield the original function inside the integral sign. These integrals are expressed as \(\int f(x) dx = F(x) + C\), where \(F(x)\) is the antiderivative of \(f(x)\), and \(C\) is the constant of integration. This constant appears because differentiation wipes out any constant term, meaning many functions can have the same derivative.

Indefinite integrals are vital for understanding how to reconstruct a function from its rate of change, essentially reversing differentiation. When verifying an indefinite integral, we differentiate the result and check that it simplifies back to the original function being integrated. Our exercise showcased this verification process, ensuring that the integral calculation is correct.

Differentiating Functions

Differentiation is a mathematical operation used in calculus to find the rate at which a function is changing at any given point. It involves taking a function \(f(x)\) and finding its derivative \(f'(x)\), which gives the slope of the tangent line to the function's graph at any point. The process of differentiating a function can involve applying various rules, including the power rule, product rule, quotient rule, and, as seen in our exercise, the chain rule.

When differentiating functions, it's important to identify whether the function is simple (directly apply basic differentiation rules) or composite (requiring the chain rule). For the function \(f(x) = \sqrt{x^2 + 1} + C\), we apply the chain rule since \(x^2 + 1\) is nested inside the square root. Properly differentiating functions allows us to verify solutions to indefinite integrals by ensuring the derivative of the result matches the original function inside the integral.

Square Root Differentiation

Differentiating functions with square roots involves treating the square root as an exponent. For instance, the square root of \(u\) is the same as \(u^{1/2}\). By applying the chain rule, we can break down the process of differentiating functions with square roots into simpler steps. The typical approach is to differentiate the function as if you were dealing with a rational exponent and then multiply by the derivative of the expression inside the square root.

In the context of our exercise, where we differentiated \(\sqrt{x^2 + 1} + C\), we initially treated the square root as the outer function and \(x^2 + 1\) as the inner function. We then differentiated \(\sqrt{u}\) as \(u^{1/2}\) and applied the power rule, combining it with the derivative of the inner function \(x^2 + 1\). The result was then simplified to match the integrand, verifying the correct application of square root differentiation and the correctness of the indefinite integral.