I am doing as many trig integrals exercices as I can to develop the skills and thought patterns. I have a rather simple question about the result I get when computing integrals such as $\int \sin^7 \theta\cos^5\theta\, d\theta$, I use the basic substitution technique
$$\int \sin^7 \theta\cos^5\theta\, d\theta\,=\,\int u^7\theta\left(1-u^2\theta\right)^2\, du\,:\,u=\sin\theta \quad du=\cos\theta\, d\theta$$ wich gives me $$\frac 18 \sin^8\theta - \frac 15 \sin^{10}\theta + \frac 1{12} \sin^{12}\theta +C$$ I check my answers with $Mathematica$ which gives, $$\frac {-5\cos{2\theta}}{1024}+\frac {5\cos{4\theta}}{8192}+\frac {5\cos{6\theta}}{6144} - \frac {\cos{8\theta}}{4096} - \frac {\cos{10\theta}}{10240}+\frac {\cos{12\theta}}{24 576}$$ this is far from the result I got by substitution so I thought I was doing something wrong and I tried to achieve the same result but it is a bit painful, so I tried a definite integral as a quicker way to convince myself. Sure enough, I get the same answer as $Mathematica$ when I compute definite integrals. For example, $$\int_0^\frac \pi 2 \sin^7 \theta\cos^5\theta\, d\theta\,=\,\frac1{120}$$ $$\frac 18 \sin^8\frac \pi 2- \frac 15 \sin^{10}\frac \pi2+ \frac 1{12} \sin^{12}\frac \pi2\,=\,\frac 1{120}$$ Does this have to do with the way $Mathematica$ does computations or am I doing something wrong? Also, is it useful to get it down to the same form as does $Mathematica$ or is the simple u-substitution result always reliable and correct?