Writing the integral $\int\limits_{\pi}^{N \pi} \frac{|sin \theta|}{|\theta|} d\theta $ as sum.

Question

Write the integral $$\int_{\pi}^{N \pi} \frac{|sin \theta|}{|\theta|} d\theta $$ as a sum $$\sum_{k = 1}^{N-1} \int_{k \pi}^{(k+1)\pi}$$

Could anyone give me a hint?

Answer 1

Using the fact $$\int_a^b+\int_b^c=\int_a^c$$ we get $$\int_{\pi}^{N \pi} \frac{|sin \theta|}{|\theta|} d\theta=\sum_{k = 1}^{N-1} \int_{k \pi}^{(k+1)\pi} \frac{|sin \theta|}{|\theta|} d\theta$$