Question

Find the integral. Use a computer algebra system to confirm your result. $$ \int \csc ^{4} \theta d \theta $$

Short answer

The integral of \(\csc^{4} \theta d \theta\) is \(-\cot \theta - \frac{2}{3}\cot^{3} \theta - \frac{1}{5}\cot^{5} \theta + C\).

Step-by-step solution

Break down the function

Write the integrand \(\csc^{4} \theta\) as \(\csc^{2} \theta * \csc^{2} \theta\), which can ultimately be expressed as \((1 + \cot^{2} \theta)*(1 + \cot^{2} \theta)\), using the trigonometric identity \(1 + \cot^{2} \theta = \csc^{2} \theta\).

Expand and Simplify

Expand the product to get \(1 + 2\cot^{2}\theta + \cot^{4}\theta\). Now, substitute \(u = \cot \theta\) and \(du = -\csc^{2} \theta d\theta\). This substitution changes the integral into \(-\int (1 + 2u^{2} + u^{4}) du\).

Integrate

Perform the integration using standard rules of integration to get \(-u - \frac{2}{3}u^{3} - \frac{1}{5}u^{5} + C\).

Substitute Back

Substitute \(u = \cot\theta\) back into the equation to get the final answer, which is \(-\cot \theta - \frac{2}{3}\cot^{3} \theta - \frac{1}{5}\cot^{5} \theta + C\).