struggling with the following question:

Evaluate the following convolution integral


any help would be much appreciated, the hint of using the cos trig identity has been given.


The hint of using trig identities is probably as follows $$\cos x*\cos x=\int_{\text{limits}^*}\cos (u')\cdot\cos(u-u')\,du'\\=\int_{\text{limits}}\cos u'(\cos u'\cos u+\sin u'\sin u)\,du'\\=\cos u\int_{\text{limits}}\cos^2u'\,du'+\sin u\int_{\text{limits}}\sin u'\cos u'\,du'$$ which can both be evaluated using identities relating to $\cos2u'$ and $\sin2u'$.

$*$ - I have written $\text{limits}$ since I'm not sure what regions you are defining $\cos u'$ on, so I suppose you can put this in.

  • $\begingroup$ your first line should it be cos(u) not cos(u') $\endgroup$ May 25 '17 at 12:31
  • $\begingroup$ @Georgewall No, I am using $u'$ as the variable with respect to which I integrated (I did have $u$ and $u'$ the wrong way around in the other term, so understand the confusion. However it did not make any difference since $\cos$ is even) $\endgroup$
    – John Doe
    May 25 '17 at 12:40

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