Questions: Evaluate the integral: [ int(1+sin^2 theta csc theta) d theta ]

Solution Steps

To evaluate the integral \(\int\left(1+\sin ^{2} \theta \csc \theta\right) d \theta\), we first simplify the integrand. Notice that \(\csc \theta = \frac{1}{\sin \theta}\), so \(\sin^2 \theta \csc \theta = \sin \theta\). Thus, the integrand simplifies to \(1 + \sin \theta\). We can then integrate each term separately.

We start with the integral

\[ \int\left(1+\sin ^{2} \theta \csc \theta\right) d \theta. \]

Recognizing that \(\csc \theta = \frac{1}{\sin \theta}\), we can rewrite \(\sin^2 \theta \csc \theta\) as \(\sin \theta\). Thus, the integrand simplifies to

\[ 1 + \sin \theta. \]

Now we can integrate the simplified expression:

\[ \int (1 + \sin \theta) d \theta. \]

This can be separated into two integrals:

\[ \int 1 \, d\theta + \int \sin \theta \, d\theta. \]

The first integral evaluates to \(\theta\), and the second integral evaluates to \(-\cos \theta\). Therefore, we have:

\[ \int (1 + \sin \theta) d \theta = \theta - \cos \theta + C, \]

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

Final Answer