Question

Describe the method used to integrate \(\sin ^{3} x\)

Short answer

Answer: The integral of sin^3(x) with respect to x is -cos(x) + (1/3)cos^3(x) + C.

Step-by-step solution

Use the trigonometric identity

Replace \(\sin^3 x\) with \((\sin x)(\sin^2 x)\) and then use the identity \(\sin^2 x = 1 - \cos^2 x\): $$ \int \sin^3 x \, dx = \int \sin x (1 - \cos^2 x) \, dx $$

Expand the integrand

Distribute \(\sin x\) to both terms inside the parentheses: $$ \int \sin x (1 - \cos^2 x) \, dx = \int (\sin x - \sin x \cos^2 x) \, dx $$

Separate into two integrals

Break the integral up into two separate terms: $$ \int (\sin x - \sin x \cos^2 x) \, dx = \int \sin x \, dx - \int \sin x \cos^2 x \, dx $$

Integrate \(\sin x\)

Integrate the first term in the expression: $$ \int \sin x \, dx = -\cos x $$

Integrate \(\sin x \cos^2 x\)

To integrate the second term, use substitution method with \(u = \cos x\) and \(du = -\sin x \, dx\): $$ \int \sin x \cos^2 x \, dx = -\int u^2 \, du $$ Now, integrate \(-\int u^2 \, du\): $$ -\int u^2 \, du = -\frac{1}{3} u^3 + C = -\frac{1}{3} \cos^3 x + C $$

Combine the results

Combine the integrals of both terms to find the integral of the original function: $$ \int \sin^3 x \, dx =(-\cos x) - (-\frac{1}{3} \cos^3 x + C) = -\cos x + \frac{1}{3}\cos^3 x + C $$ The integral of \(\sin^3 x\) is \(-\cos x + \frac{1}{3}\cos^3 x + C\).

Key concepts

Trigonometric Identities

Understanding trigonometric identities is crucial when dealing with integrals involving trigonometric functions. Trigonometric identities simplify the integrands and make them more manageable for integration. A fundamental identity used in the integration of powers of sine and cosine functions is the Pythagorean identity, which states that \(\sin^2 x + \cos^2 x = 1\). This identity allows us to convert between sine and cosine functions.

For example, in the case of \(\sin^3 x\), we use the identity to express \(\sin^2 x\) in terms of \(\cos x\). Since \(\sin^2 x = 1 - \cos^2 x\), it helps us rewrite \(\sin^3 x\) as \(\sin x(1 - \cos^2 x)\), thus breaking it down to easier functions to integrate. The application of trigonometric identities paves the way for the integration process and is often the first step in solving complex integrals.

Integration by Substitution

Integration by substitution is a technique often employed in integral calculus to simplify an integral that is difficult to evaluate directly. Commonly referred to as the reverse chain rule, it involves replacing a portion of the integrand with a new variable. This step usually simplifies the integral into a more recognizable form that can be easily integrated.

When integrating \(\sin^3 x\), after applying trigonometric identities, one of the resulting terms is \(\sin x \cos^2 x\). To integrate this term, we can use substitution by letting \(u = \cos x\) and \(du = -\sin x \, dx\). When we substitute these into the integral, we obtain a simpler integral involving \(u\) that can be readily integrated. This method is widely used because it transforms complex expressions into more solvable problems.

Integral Calculus

Integral calculus, one of the two main branches of calculus, focuses on finding the total size or value, such as areas under curves, by integrating the rate of change. Integrals can be categorized into indefinite and definite types, with the former being used to find antiderivatives and the latter calculating the actual total size between bounds.

Indefinite integrals come with an arbitrary constant of integration, denoted by \(C\), because the process of differentiation removes constants. The process involves many techniques such as substitution, integration by parts, partial fractions, and trigonometric identities. Grasping these concepts is essential when tackling integrals of functions including trigonometric functions like in \(\sin^3 x\).

U-Substitution Method

The u-substitution method is a specific type of integration by substitution that assists in solving integrals more easily. It involves choosing a part of the integrand to be \(u\) which, when replaced, simplifies the integral. It's especially helpful when dealing with products or compositions of functions.

In our example with \(\sin^3 x\), we used u-substitution when we came across the \(\sin x \cos^2 x\) term. Selecting \(u = \cos x\) and expressed \(dx\) in terms of \(du\), changed the difficult integral into a simple power function of \(u\) that is much easier to integrate. This method is widely praised for its ability to make complex integrals manageable and is a quintessential tool in a student's calculus toolbox.