Problems

Integrate the expression $$\frac{x^3}{(x^2 + 1)^3}$$

1. by using the substitution \(y = x^2 + 1\), and
2. by using the substitution \(\tan \theta = x\).

Verify that your answers are equivalent.

Show that $$\frac{dv}{du} - \frac{nv}{u} = u^n \frac{d}{du}(vu^{-n}).$$ By considering \(x\) as a function of \(y\), or otherwise, find and sketch the solution of the differential equation $$\frac{dy}{dx} = \frac{y}{3x - 2y},$$ which passes through the point \(x = 0\), \(y = 1\).

1. Evaluate the indefinite integrals
   1. Show that $$\int_{-\pi/4}^0 \frac{dx}{\cos x - \sin x} = \frac{1}{\sqrt{2}} \ln(\sqrt{2} + 1).$$

The value of \(y\) is given by \(y = a + c \ln y\), where \(c\) is small. Show that \(y\) is given approximately by $$y = a + c \ln a + \frac{c^2}{a} \ln a$$ and find the term in \(c^3\).

If $$I_n = \int_0^{\pi/2} \cos^n \theta \, d\theta,$$ find a recurrence relation for \(I_n\) and deduce that $$I_n I_{n-1} = \frac{\pi}{2n}$$ for all integers \(n > 1\).

1. Show that \((1 + t)(1 - t + t^2 + \ldots + (-1)^n t^n) = 1 + (-1)^n t^{n+1}\).
2. Using this result for \(n = 1\) show that $$t - \frac{1}{2}t^2 < \ln(1 + t) < t \quad \text{for } t > 0.$$
3. Prove that $$\left|\ln(1 + t) - \sum_{r=1}^n (-1)^{r+1} \frac{t^r}{r}\right| < \frac{t^{n+1}}{n+1} \quad \text{for } t > 0.$$