Problem 34 - Essential Calculus

Question

Find the indefinite integral. $$ \int \cos 6 x d x $$

Step-by-Step Solution

Answer

The indefinite integral of \( \cos 6x \) is \( \frac{1}{6}\sin 6x + C \).

Step 1: Identify the integrand

The integrand (the function to be integrated) in this case is \(\cos 6x\). In this function, \(6x\) is the argument of the \(\cos\) function, so \(a = 6\).

Step 2: Apply the integration rule

Applying the integration rule \(\int \cos ax \, dx = \frac{1}{a}\sin ax + C\), the indefinite integral of the given function can be obtained. Here, \(a = 6\). So, the function becomes \(\frac{1}{6}\sin 6x\).

Step 3: Add the constant of integration

Finally, the constant of integration 'C' is added to the result. This is because the derivative of any constant is 0, so when we are finding anti-derivatives, we add this constant. Thus, the final solution to the indefinite integral of the given function is \(\frac{1}{6}\sin 6x + C\).