Question

In Exercises \(25-46,\) use substitution to evaluate the integral. $$\int 3(\sin x)^{-2} d x$$

Short answer

The evaluated integral of \(3(\sin x)^{-2} dx\) is \(-3\cot{x} + C\).

Step-by-step solution

Write the original integral in a simplified form

Let's transform the expression under the integral to a form better suited for substitution. The original integral, \[\int 3(\sin x)^{-2} dx ,\] can be rewritten as \[\int 3 \cdot \csc^2{x} dx .\] This is a more familiar form with potential for substitution.

Make a substitution

Next, make a substitution with \(u = \cot{x}\). The derivative of \(\cot{x}\) is \(-\csc^2{x}\), therefore \(du = -\csc^2{x} dx\). Now, substitute back into the original integral to get \[-3 \int du .\]

Calculate the integral with respect to \(u\)

Now, we have a simple integral to solve. \[-3 \int du = -3u + C,\] where \(C\) is the constant of integration.

Substitute \(\cot{x}\) back for \(u\)

Finally, replace \(u\) with \(\cot{x}\) to get the final solution \[-3u + C = -3\cot{x} + C .\] This is the evaluated integral.