Problem 35 - Essential Calculus

Question

Find the indefinite integral. $$ \int \frac{1}{\theta^{2}} \cos \frac{1}{\theta} d \theta $$

Step-by-Step Solution

Answer

The indefinite integral is \( -\sin \left( \frac{1}{\theta} \right) + C \)

Step 1: Identify and Perform a u-Substitution

Firstly, identify that the integrand can be simplified through u-substitution. Letting \( u = \frac{1}{\theta} \) will simplify the integral considerably. Don't forget to find \(du\), which in this case, is \( du = - \frac{1}{\theta^{2}} d\theta \). After the substitution, the integral becomes \( -\int \cos(u) du \).

Step 2: Integrate the Simplified Function

The integral should now be a standard integral that can be computed directly. The integral of cosine function is the sine function, that will make the integral become \( -\sin(u) + C \) where \( C \) is the constant of integration.

Step 3: Substituting u Back

Finally, replace \(u\) with the original expression, \( \frac{1}{\theta} \), to get the solution in terms of \( \theta \). Then the final solution is: \( -\sin \left( \frac{1}{\theta} \right) + C \).