Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best viewed in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (you should be able to scroll/swipe to see them) and some of the menu items will be cut off due to the narrow screen width.
Section 7.2 : Integrals Involving Trig Functions
9. Evaluate \( \displaystyle \int_{1}^{3}{{\sin \left( {8x} \right)\sin \left( x \right)\,dx}}\).
Show All Steps Hide All Steps
Start SolutionThere really isn’t all that much to this problem. All we have to do is use the formula given in this section for reducing a product of a sine and a cosine into a sum. Doing this gives,
\[ \int_{1}^{3}{{\sin \left( {8x} \right)\sin \left( x \right)\,dx}} = \int_{1}^{3}{{\frac{1}{2}\left[ {\cos \left( {8x - x} \right) - \cos \left( {8x + x} \right)} \right]\,dx}} = \frac{1}{2}\int_{1}^{3}{{\cos \left( {7x} \right) - \cos \left( {9x} \right)\,dx}}\] Show Step 2Now all we need to do is evaluate the integral.
\[\begin{align*}\int_{1}^{3}{{\sin \left( {8x} \right)\sin \left( x \right)\,dx}} & = \left. {\frac{1}{2}\left[ {\frac{1}{7}\sin \left( {7x} \right) - \frac{1}{9}\sin \left( {9x} \right)} \right]} \right|_1^3\\ & = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{1}{{14}}\sin \left( {21} \right) - \frac{1}{{18}}\sin \left( {27} \right) - \frac{1}{{14}}\sin \left( 7 \right) + \frac{1}{{18}}\sin \left( 9 \right) = - 0.0174}}\end{align*}\]Make sure your calculator is set to radians if you computed a decimal answer!