I tried it using the double angle identity
$$\sin{2x}=2\sin x\cos x$$
The answer that I got is
$$\frac{-\cos 2x}{4} +c$$
However I've also tried it using $u$-substitution.
I let $u=\sin x$. Thus obtaining $\cos x$ when differentiating. And cutting the $\cos x$ in $2\sin x\cos x$ out with the $\cos x$ in the denominator below $du$.
However the answer that I am then getting is : $0.25 – 0.25\cos 2x + c$. So as you can see there is the extra term $0.25$ there. Is the second answer deemed to be wrong? If so why? My book tells me to use the double angle formula but does not explain why.
Best Answer
Both answers are correct. You can use double angle identity, as well as u sub for either $\sin x$ or $\cos x$.
The key lies in the +c. All the 3 integrals are a family of functions just separated by a different "+c". In practice, double angle identity is often used as it's more intuitive and simpler in some sense. But the other methods are perfectly acceptable, and not "wrong."