Problem 58 - Calculus Single Variable

Question

The denominator of the given integrand is of the form \(a \pm b\). Multiply numerator and denominator by \(a \mp b\) to obtain a difference of squares in the denominator. Then use an appropriate trigonometric identity before integrating. $$ \int \frac{1}{1+\cos (x)} d x $$

Step-by-Step Solution

Answer

The integral evaluates to \(-\cot(x) - \csc(x) + C\).

Step 1: Identify the form of the denominator

The denominator of the integrand is \(1 + \cos(x)\). According to the exercise, we need to multiply both numerator and denominator by its conjugate, which is \(1 - \cos(x)\), to form a difference of squares.

Step 2: Multiply integrand by conjugate over itself

Multiply the integrand \(\frac{1}{1+\cos(x)}\) by \(\frac{1-\cos(x)}{1-\cos(x)}\), resulting in: \[ \int \frac{1-\cos(x)}{(1+\cos(x))(1-\cos(x))} \, dx \] This equals \(\int \frac{1-\cos(x)}{1 - \cos^2(x)} \, dx \) because the denominator forms a difference of squares.

Step 3: Simplify using a trigonometric identity

The difference of squares \(1 - \cos^2(x)\) is equivalent to \(\sin^2(x)\) by the identity \(\sin^2(x) + \cos^2(x) = 1\). Thus, the integral becomes: \[ \int \frac{1-\cos(x)}{\sin^2(x)} \, dx \] Now split it into two separate integrals: \[ \int \frac{1}{\sin^2(x)} \, dx - \int \frac{\cos(x)}{\sin^2(x)} \, dx \]

Step 4: Simplify integrals with trigonometric identities

Remember that \(\frac{1}{\sin^2(x)} = \csc^2(x)\) and \(\frac{\cos(x)}{\sin^2(x)} = \cot(x) \csc(x)\). Thus, the integrals become: \[ \int \csc^2(x) \, dx - \int \cot(x) \csc(x) \, dx \] The integral of \(\csc^2(x)\) is \(-\cot(x)\), and the integral of \(\cot(x) \csc(x)\) is \(\csc(x)\).

Step 5: Integrate and find the final result

Perform the integration to find: \[ -\cot(x) - \csc(x) + C \] where \(C\) is the constant of integration. This completes the integration of the function.

Key Concepts

Difference of Squares

The difference of squares is a powerful algebraic tool used to simplify expressions. It takes the form \(a^2 - b^2\) and can be factored as \((a+b)(a-b)\). In our integral problem, the denominator is given as \(1 + \cos(x)\).
To leverage the difference of squares, we multiply this by its conjugate, \(1 - \cos(x)\). This operation results in:

• \((1+\cos(x))(1-\cos(x)) = 1 - \cos^2(x)\)

This simplifies the expression significantly.
By doing this, we transform the denominator into a form that is easier to work with, setting the stage for applying trigonometric identities to continue simplifying our integral expression.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for any value of the variable involved. These identities help simplify complex trigonometric expressions. For the problem at hand, once we reformulate the denominator as \(1 - \cos^2(x)\), we can use the Pythagorean identity:

• \(\sin^2(x) + \cos^2(x) = 1\)

This rearranges to show:

• \(1 - \cos^2(x) = \sin^2(x)\)

This transformation allows us to rewrite the integral in terms of \(\sin(x)\), streamlining the integration process. Considering the identity, we get:

• \(\int \frac{1-\cos(x)}{\sin^2(x)} \, dx\)

This sets up a nice segue into separation, breaking the fraction into simpler parts that can be individually integrated using more basic trigonometric identities.

Integration Techniques

Integration techniques involve the methods used to find the integral of a function, which includes substitution, integration by parts, and recognizing standard integral forms. Here, we make use of known integrations involving trigonometric functions:

• \(\int \csc^2(x) \, dx = -\cot(x) + C\)
• \(\int \cot(x) \csc(x) \, dx = -\csc(x) + C\)

The integral initially appears as:

• \(\int \frac{1}{\sin^2(x)} \, dx - \int \frac{\cos(x)}{\sin^2(x)} \, dx\)

By recognizing that \(\frac{1}{\sin^2(x)}\) equals \(\csc^2(x)\), and \(\frac{\cos(x)}{\sin^2(x)}\) equals \(\cot(x) \csc(x)\), we're able to apply these standard forms directly.
This simplifies the integral to:

• \(\int \csc^2(x) \, dx - \int \cot(x) \csc(x) \, dx\)

The final result beautifully combines these techniques and identities to yield:

• \(-\cot(x) - \csc(x) + C\)

Here, \(C\) represents the constant of integration, signifying that this is a general solution.