Problem 50 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises \(47-52,\) use the given trigonometric identity to set up a \(u\) -substitution and then evaluate the indefinite integral. $$\int 4 \cos ^{2} x d x, \quad \cos 2 x=1-2 \cos ^{2} x$$

Step-by-Step Solution

Answer

The integral of \(4 \cos ^{2} x dx\) is \(2x - \sin 2x + C\).

Step 1: Apply the trigonometric identity

First, expressing \(\cos ^{2} x\) in terms of \(\cos 2x\) using the given trigonometric identity, we get \(\cos ^{2} x = (1-\cos 2x)/2\). This simplifies the integral to \( \int 4 (1-\cos 2x)/2 dx= \int 2 (1-\cos 2x) dx\)

Step 2: Break up the integral

Split this into two separate integrals. The integral of a sum is equal to the sum of the integrals. Thus, it can be written as: \( \int 2 dx - \int 2 \cos 2x dx\).

Step 3: Reformulate using u-substitution

For the second integral, perform a \(u\)-substitution. Let \(u = 2x\). Therefore, \(du = 2 dx\). Applying this substitution, the integral become \(- \int \cos u du\).

Step 4: Evaluate the Integrals

Evaluate each integral. \(\int 2 dx = 2x\), and \( -\int \cos u du = - \sin u\).

Step 5: Substitute u back in terms of x

Replace u back into the equation with the original variable, \(2x\). This results in \(- \sin 2x\).

Step 6: Combine the results

Combine these results to get the original result. The computed integral is: \(2x - \sin 2x + C\), where C is the constant of integration.

Key Concepts

Indefinite Integral

In calculus, the concept of an indefinite integral is crucial. It represents the family of all antiderivatives of a function. When you integrate a function without limits, you obtain an indefinite integral.
This process is the reverse operation of differentiation.

• The result of an indefinite integral is a general solution to the problem of finding a function whose derivative is the given function.
• It is commonly expressed as \( \int f(x) \, dx \), where \( f(x) \) is the function being integrated.
• A constant of integration, often denoted as \( C \), is added because there are infinitely many possible antiderivatives that differ by a constant factor.

When solving an indefinite integral, we aim to find a function whose rate of change is described by the integrand. This process helps deepen understanding of how functions accumulate values over an interval.

Trigonometric Identity

Trigonometric identities are equations that are true for all values of the angles involved. They are incredibly useful in simplifying complex trigonometric expressions and can help in integration and differentiation tasks.
In this exercise, we used the identity \( \cos 2x = 1 - 2\cos^2 x \) to simplify an integral involving \( \cos^2 x \).

• This identity expresses \( \cos^2 x \) in terms of \( \cos 2x \), which can simplify the integration process.
• Using identities allows converting complex integrals into simpler forms that are easier to evaluate.
• Trigonometric identities are tools in calculus that help manipulate integrands into integrable forms.

Mastering these identities enables us to recognize patterns and choose appropriate strategies for integration, such as converting squared trigonometric functions into linear ones.

Integration Techniques

Integration techniques are methods used to evaluate integrals. One of the essential techniques is the substitution method, often called \( u \)-substitution. This technique simplifies an integral by making a substitution that transforms the integral into an easier one.For the given problem, after simplifying using the trigonometric identity, we make a \( u \)-substitution for the integral part involving \( \cos 2x \):

• Choose \( u = 2x \). This is often done when the derivative of \( u \) appears in the integral.
• The derivative \( du = 2 \, dx \) helps rewrite the differential part of the integral.
• The integral transforms into an easier form to solve, such as \(-\int \cos u \, du \).

Integration techniques like substitution are fundamental to solving complex integrals by breaking them into simpler forms. Knowing when and how to apply these techniques is vital for efficient problem-solving in calculus.