Evaluate the following integral. ∫ sin 5 x d x

This integral no longer has the cosine in it that would allow us to use the substitution that we used above. Therefore, that substitution won’t work and we are going to have to find another way of doing this integral.

Let’s first notice that we could write the integral as follows,

∫sin5xdx=∫sin4xsinxdx=∫(sin2x)2sinxdx

Now recall the trig identity

$${\cos }^{2}x+{\sin }^{2}x=1\Rightarrow {\sin }^{2}x=1-{\cos }^{2}x$$

With this identity the integral can be written as,

$$\mathrm{and\; we\; can\; now\; use\; the\; substitution u=cosxu=cosx.\; Doing\; this\; gives\; u}$$

$$\mathrm{\int sin5xdx=-\int (1-u2)}$$