(Solved): The three basic trigonometric substitutions are in the table below. Using the substitution: \( x=1 ...

The three basic trigonometric substitutions are in the table below. Using the substitution: \( x=1 \sec \theta, 0<\theta<\frac{\pi}{2} \), re-write the indefinite integral then evaluate in terms of \( \theta \). Suppose that \( x>1 \). Then, \[ \int \frac{1}{x^{4} \sqrt{x^{2}-1}} d x=\int= \] Note: type 'theta' for \( \theta \) and 'dtheta' for \( d \theta \). Part \( 2 . \) Back substituting in the antiderivative you found in Part 1. above we have \[ \int \frac{1}{x^{4} \sqrt{x^{2}-1}} d x=\frac{\sqrt{x^{2}-1}}{x}-\frac{\left(x^{2}-1\right) \sqrt{x^{2}-1}}{3 x^{3}}+C \] Note: answer should be in terms of \( x \) only.