# laplace transform using convolution intergral

struggling with the following question:

Evaluate the following convolution integral

cos(x)*cos(x)

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

## 1 Answer

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.

• your first line should it be cos(u) not cos(u') May 25 '17 at 12:31
• @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) May 25 '17 at 12:40