Question

Integral of mod sinx from pi to 10pi

Answer 1

Given:

• A function  $\int_0^{10\pi}|\sin x|dx&$ is given

To Find:

• Integral of  $\int_0^{10\pi}|\sin x|dx&$

Solution:

We know that,

            $sin(x+\pi)= -sin(x)$

So,

     $|sin(x+\pi)|=|sin(x)|$ It  means $|sinx|$  is periodic, with period  $\pi$ .

In any periodic function with period  T

          $\int\limits^{a+kT}_a {f(x)} dx = k \int\limits^{a+T}_a f(x)dx$

         $\int_0^{10\pi}|\sin x|dx&=10\int_0^\pi|\sin x|dx$

                               $&=10\int_0^\pi\sin x\;dx$

                               $&=10\bigl(-\cos x\Bigr|_0^\pi$

                               $&=10(-\cos\pi+\cos0)$

                               $&=10\times2\\&=20$

So,

            $\begin{equation}\boxed{\int_0^{10\pi}|\sin x|dx=20}$