Questions: Find the indefinite integral. (Remember the constant of integration.) ∫ 3/θ^2 cos(3/θ) dθ

Solution Steps

The given integral is:

\[ \int \frac{3}{\theta^{2}} \cos \left(\frac{3}{\theta}\right) d \theta \]

This is a composite function, and it suggests the use of substitution to simplify the integration process.

Let's choose the substitution:

\[ u = \frac{3}{\theta} \]

Then, differentiate \( u \) with respect to \( \theta \):

\[ \frac{du}{d\theta} = -\frac{3}{\theta^2} \]

This implies:

\[ d\theta = -\frac{\theta^2}{3} \, du \]

Substitute \( u = \frac{3}{\theta} \) and \( d\theta = -\frac{\theta^2}{3} \, du \) into the integral:

\[ \int \frac{3}{\theta^{2}} \cos \left(\frac{3}{\theta}\right) d \theta = \int \frac{3}{\theta^2} \cos(u) \left(-\frac{\theta^2}{3}\right) du \]

Simplify the expression:

\[ = \int -\cos(u) \, du \]

The integral of \(-\cos(u)\) with respect to \(u\) is:

\[ \int -\cos(u) \, du = -\sin(u) + C \]

where \(C\) is the constant of integration.

Replace \(u\) with \(\frac{3}{\theta}\) to express the result in terms of \(\theta\):

\[ -\sin\left(\frac{3}{\theta}\right) + C \]

Final Answer