Chapter 2

For the following exercises, determine whether the statement is true or false. Justify your answer with a proof or a counterexample\(\int \frac{1}{x^{4}+1} d x\) cannot be integrated using partial fractions.

For the following exercises, solve the differential equations. \(\frac{d y}{d \theta}=\sin ^{4}(\pi \theta)\)

For the following problems, use the substitutions \(\tan \left(\frac{x}{2}\right)=t, d x=\frac{2}{1+t^{2}} d t, \sin x=\frac{2 t}{1+t^{2}}\), and \(\cos x=\frac{1-t^{2}}{1+t^{2}}\). Find the area under the curve \(y=\frac{1}{1+\sin x}\) between \(x=0\) and \(x=\pi\). (Assume the dimensions are in inches.)

Find the length of the curve \(y=\ln (\csc x), \frac{\pi}{4} \leq x \leq \frac{\pi}{2}\)

For the following problems, use the substitutions \(\tan \left(\frac{x}{2}\right)=t, d x=\frac{2}{1+t^{2}} d t, \sin x=\frac{2 t}{1+t^{2}}\), and \(\cos x=\frac{1-t^{2}}{1+t^{2}}\). Given \(\tan \left(\frac{x}{2}\right)=t\), derive the formulas \(d x=\frac{2}{1+t^{2}} d t, \sin x=\frac{2 t}{1+t^{2}}\), and \(\cos x=\frac{1-t^{2}}{1+t^{2}} .\)

For the following exercises, determine whether the statement is true or false. Justify your answer with a proof or a counterexampleIn numerical integration, increasing the number of points decreases the error.

Find the length of the curve \(y=\ln (\sin x), \frac{\pi}{3} \leq x \leq \frac{\pi}{2}\).

For the following exercises, evaluate the integral using the specified method.\(\int x^{2} \sin (4 x) d x\) using integration by parts

Find the volume generated by revolving the curve \(y=\cos (3 x)\) about the \(x\) -axis, \(0 \leq x \leq \frac{\pi}{36}\).

For the following exercises, use this information: The inner product of two functions \(f\) and \(g\) over \([a, b]\) is defined by \(f(x) \cdot g(x)=\langle f, g\rangle=\int_{a}^{b} f \cdot g d x\). Two distinct functions \(f\) and \(g\) are said to be orthogonal if \(\langle f, g\rangle=0\). Show that \(\\{\sin (2 x), \cos (3 x)\\}\) are orthogonal over the interval \([-\pi, \pi]\).