Problem 64 - Essential Calculus

Question

Find the integral by using the appropriate formula. $$ \int x^{2} \cos x d x $$

Step-by-Step Solution

Answer

The solution to \(\int x^{2} \cos x dx\) is \(x^2 \sin(x) +2x \cos(x) - 2 \sin(x) + C\).

Step 1: Choose 'u' and 'dv'

Firstly, denote part of the integrand as 'u' and the other part as 'dv'. In this case, \(u = x^2\) and \(dv = \cos(x) dx\). This choice is made because the derivative of \(x^2\) (which is '2x'), becomes simpler, and the antiderivative of \(\cos(x)\) can be easily found.

Step 2: Find 'du' and 'v'

Find the derivative of 'u' and the antiderivative of 'dv'. The derivative of \(u = x^2\) is \(du = 2x dx\), and the antiderivative of \(\cos(x) dx\) (which is 'dv') is \(v = \sin(x)\).

Step 3: Apply integration by parts formula

Substitute back into the integration by parts formula to get \( \int x^2 \cos(x) dx = x^2 \sin(x) - \int 2x \sin(x) dx\)

Step 4: Solve the remaining integral

The remaining integral can also be solved by parts, again choosing \(u = 2x\) and \(dv = \sin(x) dx \). After differentiating and integrating respectively we get \( du = 2dx \) and \( v = -\cos(x) \). Now filling those into our integral we get: \( \int 2x \sin(x) dx = -2x\cos(x) - \int -2\cos(x) dx\) = -2x\cos(x) +2\sin(x).\n Substitute this back into the equation resulted from step 3: \( \int x^2 \cos(x) dx = x^2 \sin(x) - (-2x\cos(x) +2\sin(x))\), which simplifies to \(x^2 \sin(x) +2x \cos(x) - 2 \sin(x) \).

Step 5: Final solution

Don't forget to add the constant of integration (+ C) at the end. So the final answer is \(x^2 \sin(x) +2x \cos(x) - 2 \sin(x) + C\).

Key Concepts

Indefinite Integrals

An indefinite integral is a type of integral that does not have specific limits of integration. Instead, it represents a family of functions. When you see an integral symbol like \(\int\), without limits, it means you're being asked to find the indefinite integral. This process is also known as finding the antiderivative of a function.
In the expression \(\int x^{2} \cos x \, dx\), the main task is to find a function whose derivative will give you the original function \(x^{2} \cos x\).
Unlike definite integrals, indefinite integrals include an arbitrary constant, often denoted as \(+ C\), since differentiation of constants results in zero.
This constant reminds us that there could be many functions with different constant terms that share the same derivative.

• Indefinite integrals don't set boundaries or intervals.
• The process finds a general form of the antiderivative.
• Always add \(+ C\) as the integration constant.

Understanding indefinite integrals helps us grasp fundamental concepts in calculus, such as antiderivatives and integration by parts.

Antiderivatives

Antiderivatives are functions that "reverse" the process of differentiation. If the derivative of a function \(F(x)\) is \(f(x)\), then \(F(x)\) is an antiderivative of \(f(x)\). The entire process of finding an antiderivative is equivalent to performing an indefinite integral.
Finding antiderivatives can sometimes be straightforward, especially with basic polynomials or exponential/trigonometric functions. However, when dealing with products of functions or more complex expressions, techniques like integration by parts come into play. This method helps when the integral involves functions such as \(x^{2} \cos x\), where both polynomial and trigonometric parts exist.
In our exercise, recognizing that the integrand is the product of \(x^{2}\) and \(\cos x\) suggests that integration by parts will simplify the process of finding the antiderivative.

• An antiderivative represents the reverse of differentiation.
• Used in finding the function whose derivative matches the given function.
• Techniques like substitution or integration by parts may be necessary for complex expressions.

By understanding antiderivatives, we deepen our comprehension of calculus concepts, fostering problem-solving skills in different scenarios.

Trigonometric Integrals

Trigonometric integrals involve the integration of functions with trigonometric components like \(\sin x\), \(\cos x\), or \(\tan x\). These types of integrals can pose unique challenges, often requiring specific techniques or identities for simplification and solution.
In the context of our original exercise, the presence of \(\cos x\) in the integrand \(x^{2} \cos x\) necessitates identifying the antiderivative. The integral of \(\cos x\) is relatively straightforward, found to be \(\sin x\). This knowledge is pivotal during the integration by parts process.
When engaging with trigonometric integrals, it's helpful to:

• Be familiar with basic trigonometric identities and antiderivatives.
• Apply integration techniques, such as substitution or integration by parts, effectively.
• Recognize patterns or symmetries within trigonometric expressions that may simplify integration.

Understanding trigonometric integrals equips us with essential tools to tackle diverse problems, bridging connections between algebra, calculus, and trigonometry.