0

I tried finding $\int{\dfrac{\cos{x}}{2+\sin{x}}}dx$ via Weierstraß substitution: $t = \tan{\frac{x}{2}}$ and alternatively via $u = 2 + \sin{x}$.

The first approach results in a rational function in $t$, applying partial fraction decomposition and solving elementary integrals. Eventually it yields $$\log{\left(\frac{1}{2}\sin{(x)}+1\right)}$$

The second approach is more straight forward and gives

$$\log{(\sin{(x)}+2)}$$

Are these expressions potentially equal since I could not find any computation error.

spectre42
  • 301
  • It's important to keep track of delimiters. The antiderivatives should be $\log\left(\frac12\sin x+1\right)$ and $\log\left(\sin x+2\right)$. Now use the fact that $\log(ab) = \log a + \log b$ to reconcile them. – user170231 Dec 01 '23 at 18:34
  • @user170231 right, but $\log{\left(\frac{1}{2}\right)}+\log{(\sin{(x)}+2)} \neq \log{(\sin{(x)}+2)}$ or am I mistaken? – spectre42 Dec 01 '23 at 18:40
  • 1
    There's a constant of integration.... – Brevan Ellefsen Dec 01 '23 at 18:42
  • @BrevanEllefsen right, thank you. And the constant they differ is $\log{\left(\frac{1}{2}\right)}$ – spectre42 Dec 01 '23 at 18:45

0 Answers0