Integration of $\sin(2x)$ via two different methods [duplicate]

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When I integrate $\sin(2x)$ first using the trig substitution of $\sin(2x) = 2 \sin x\cos x$ and then using u-substitution I get $\sin^2(x) +C$. Integrating the same expression using u-substitution directly, I get $-\cos(2x)/2+C$.

These answers do not seem to be equivilent. When I let $x=0$ in the first example's answer I get $=0$ but when I let $x=0$ in the seccond answer I get $-1/2$.

Clearly I must be doing something wrong. If I had made these integrations definite integrals over $0$ to $\pi/2$ seems like I would have gotten different answsers.

Can you assist?

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1 Answer

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See this related post. This is a commmon mistake while doing integration.

For example, if $f(x)=g(x)$, then would you say $\displaystyle \int f(x) \mathrm dx=\int g(x) \mathrm dx$?

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