Question

Find the indefinite integral and check the result by differentiation. $$ \int \frac{4 y}{\sqrt{1+y^{2}}} d y $$

Short answer

The indefinite integral of the function \( \frac{4 y}{\sqrt{1+y^{2}}} \) is \( 4\sqrt{1+y^2} + C \).

Step-by-step solution

Identify the integral to be calculated

The integral to be calculated is \( \int \frac{4 y}{\sqrt{1+y^{2}}} d y \)

Solve the integral

As the derivative of \(1 + y^2\) is \(2y\), which appears in the numerator, it is best to make a substitution. Let \( u = 1 + y^2\), then \( du = 2y \, dy \). The integral now becomes \( \int \frac{2 du}{\sqrt{u}} \). Solving this gives \(2 \times 2\sqrt{u}\) which simplifies to \( 4\sqrt{1+y^2} + C \), where C is the constant of integration

Check the result by differentiation

Differentiate the result using the chain rule: \( \frac{d}{dy}(4\sqrt{1+y^2}) \). Differentiating gives us back the original function \( \frac{4y}{\sqrt{1+y^2}} \), thus confirming the integration was done correctly

Key concepts

Integration by Substitution

When an integral includes a function and its derivative, we can simplify the process by using integration by substitution. This is somewhat analogous to the reverse process of the chain rule in differentiation. To apply this method, we need to identify a part of the integrand that can be replaced by a single variable to make integration easier.

For example, consider the integral in our exercise, \( \frac{4 y}{\sqrt{1+y^{2}}} dy \). The expression \(1+y^{2}\) in the denominator suggests a suitable substitution because its derivative, \(2y\), appears in the numerator. By substituting \(u = 1 + y^2\), we derive \(du = 2y dy\), which simplifies the original integral into a form that is straightforward to integrate.

Integration by substitution is particularly effective for integrals involving powers and roots, trigonometric identities, or exponential functions where direct integration would be complex or not readily apparent.

Chain Rule Differentiation

The chain rule is a staple in calculus for finding the derivative of the composition of two or more functions. If we have a function \(g(f(x))\) and we want to find its derivative, we apply the chain rule which states that \(\frac{d}{dx} [g(f(x))] = g'(f(x)) \cdot f'(x)\).

In the context of checking our indefinite integral, we had to differentiate \(4\sqrt{1+y^2}\) with respect to \(y\). Here, \(g(u) = 4\sqrt{u}\) and our inner function \(f(y) = 1 + y^2\). According to the chain rule, the derivative would be the outer function's derivative, \(g'(u)\), times the derivative of the inner function, \(f'(y)\), which results in the original integrand, confirming the correctness of our integration by substitution.

Integration of Rational Functions

Integrating rational functions, which are ratios of polynomials, can sometimes be challenging, but certain types of rational functions lend themselves to specific techniques. For instance, a rational function that includes the derivative of its denominator or part of the denominator in the numerator is often a perfect candidate for integration by substitution.

The given exercise \( \frac{4 y}{\sqrt{1+y^{2}}} dy \) is an example of such a scenario. Here the integration becomes much more manageable after recognizing the numerator includes the derivative of the denominator's inner function. While this specific example doesn't involve polynomial division or partial fraction decomposition, these are other strategies that might be employed when dealing with more complex rational functions.

Understanding how to integrate rational functions is essential, as they frequently appear in real-world applications such as engineering, physics, and economics. Therefore, mastering techniques like substitution, partial fractions, and polynomial long division can provide a solid foundation for tackling these integrals.