Problem 19 - Calculus

Question

Find the indefinite integrels in \(1-20\). \(\int \cos x \tan x d x\)

Step-by-Step Solution

Answer

The solution is \( \int \cos x \tan x \, dx = -\cos x + C.\)

Step 1: Understand the Problem

We are asked to find the indefinite integral of the function \( \cos x \tan x \). It means we need to find a function whose derivative is \( \cos x \tan x \).

Step 2: Recall Trigonometric Identities

Recognize that the tangent function can be rewritten using basic trigonometric identities. Recall that \( \tan x = \frac{\sin x}{\cos x} \).

Step 3: Substitute the Trigonometric Identity

Substitute \( \tan x = \frac{\sin x}{\cos x} \) into the integral: \[ \int \cos x \tan x \, dx = \int \cos x \left( \frac{\sin x}{\cos x} \right) \, dx. \] This simplifies to \( \int \sin x \, dx \) because \( \cos x \) in the numerator and denominator cancels out.

Step 4: Solve the Simplified Integral

We now integrate \( \int \sin x \, dx \). The integral of \( \sin x \) is known to be \( -\cos x \). So, \[ \int \sin x \, dx = -\cos x + C, \] where \( C \) is the constant of integration.

Step 5: Write the Final Result

Finally, conclude the solution by writing:\[ \int \cos x \tan x \, dx = -\cos x + C. \]

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in calculus, particularly when solving integrals involving trigonometric functions. They allow us to rewrite functions in simpler forms, making integration more straightforward. One of the most commonly used identities is the identity for the tangent function. This can be expressed as

• \(\tan x = \frac{\sin x}{\cos x}\)

By using this identity, we can simplify integrals involving tangent and cosine. For instance, when asked to integrate \(\cos x \tan x\), substituting \(\tan x\) with \(\frac{\sin x}{\cos x}\) results in the expression \(\cos x \left( \frac{\sin x}{\cos x} \right)\).
This simplification leads to a much easier integral, \(\int \sin x \, dx\), because the \(\cos x\) terms cancel each other out. Understanding and effectively applying trigonometric identities can greatly ease the process of integration.

Integral of Sine

The integral of the sine function is one of the fundamental integrals in calculus. It is key to understanding basic integration techniques.
To solve \(\int \sin x \, dx\), we need to find an antiderivative—a function whose derivative is \(\sin x\).
This antiderivative is

• \(-\cos x\)

It follows that the integral of sine can be expressed as

• \(\int \sin x \, dx = -\cos x + C\)

where \(C\) is a constant of integration.Knowing the integral of sine helps in solving a wide range of practical and theoretical problems.
For many students, this pattern of integration becomes a natural and almost automatic part of calculus problem solving. It's essential to become comfortable with this and other basic integral forms, as they form the foundation of more complex integrals.

Constant of Integration

When we say "indefinite integral," it means we integrate without limits, leading to a family of functions. This is because integrals essentially perform the reverse operation of differentiation, and differentiation wipes out any constant term from a function. When we integrate a function, we add back this arbitrary constant, known as the constant of integration, denoted as \(C\). The reason for \(C\) is simple:

• When differentiating, any constant term in the original function disappears. For example, the derivative of \(x + 5\) is the same as the derivative of \(x\), which is 1.
• To account for this loss of information, we add \(C\) back when integrating, to indicate that there are infinitely many antiderivatives, differing by a constant.

In practice, always include \(C\) at the end of your indefinite integral answers.
It reminds us that any constant could have been present in the function we started with before differentiation.