1

How can I approach this problem $$\int_{-1}^{\frac32} |x\sin(nx)| dx$$

How to handle these type of integrals?

Apart from the presence of $n$ as an argument of $sine$ function, how do I even check where it is positive or negative in the given interval so as to remove the mod?

So Lo
  • 1,567
  • You need to split the integral into the sum of integrals in which the integrand of each is of one sign. – Mark Viola Jun 13 '18 at 16:50
  • I would apply taylor series of $\sin x$ to $\sin (xn)$ – Micheal Brain Hurts Jun 13 '18 at 16:52
  • @MarkViola yes but how ? – So Lo Jun 13 '18 at 17:18
  • Which is the (original) problem exactly? The integration limits are correct? – dan_fulea Jun 13 '18 at 17:29
  • This is a nuisance (and it would be trivial to edit your question for this reason), however when one is faced with fractional bounds in an integral, it is best to typeset them with a forward slash rather than with the \frac command: $$\begin{array}{c} \text{Bad} & \text{Better} \ \hline \ \displaystyle\int_{-1}^{\frac 32}|x\sin(nx)| \ dx & \displaystyle\int_{-1}^{3/2}|x\sin(nx)| \ dx \ \end{array}$$ – Crosby Jun 20 '18 at 14:18

1 Answers1

2

Here is a sketch of a way forward.

First we write

$$\begin{align} \int_{-1}^{3/2} |x\sin(nx)|\,dx&=2\int_{0}^1 x|\sin(nx)|\,dx+\int_1^{3/2} x|\sin(nx)|\,dx\\\\ &=\frac2{n^2}\int_0^n x|\sin(x)|\,dx+\frac1{n^2}\int_n^{3n/2}x|\sin(x)|\,dx \end{align}$$

Next, we note there exists an integer $k\ge 0$ such that $k\pi\le n\le (k+1)\pi$. So, given $n$ we can write

$$\begin{align} \int_0^n x|\sin(x)|\,dx&=\sum_{m=0}^{k-1} \int_{m\pi}^{(m+1)\pi}x|\sin(x)|\,dx+\int_{k\pi}^n x|\sin(x)|\,dx\\\\ &=\sum_{m=0}^{k-1}\int_0^\pi (x+m\pi)\sin(x)\,dx+\int_0^{n-k\pi} (x+k\pi)\sin(x)\,dx \tag1 \end{align}$$

The integrals on the far right-hand side of $(1)$ can be carried out in closed form using, for example, integration by parts. And the integral $\int_n^{3n/2}x|\sin(x)|\,dx$ can be evaluated in an analogous manner. The details are left to the reader.

Mark Viola
  • 179,405