\(\begin{align*} \int \frac{\sqrt{tanx}}{sin(2x)}dx \end{align*}\) |
Connaissances:
- Changement de variable
- Trigonométrie
- Trigonométrie notation anglosaxonne
\(\begin{align*} I & = \int \frac{\sqrt{tanx}}{sin(2x)}dx \end{align*}\)
Posons le changement de variable:
\(\begin{align*} & \begin{cases} u = \sqrt{tanx} \Rightarrow tan x = u² \\
sec²x.dx =2u.du \Rightarrow dx = \frac{2u.du}{sec²x} \end{cases} \\
I & = \int \frac{u}{sin(2x)}\frac{2u.du}{sec²x} \\
& = \int \frac{u}{\cancel{2}sinx.\cancel{cosx}} \times cos^{\cancel{2}}x.\cancel{2}u.du \\
& = \int \frac{u²cosx}{sinx}du \\
& = \int \frac{u²}{tanx}du \\
& = \int \frac{u²}{u²}du = \int du \\
& = u + C = \sqrt{tanx}+C
\end{align*}\)
\[\boxed {\begin{align*} I & = \sqrt{tanx}+C(\in \mathbb R) \end{align*}}\]