\(\begin{align*} \int \frac{1}{csc^3x}dx \end{align*}\) |
Connaissances:
- Trigonométrie
- Trigonométrie notation anglosaxonne
- Changement de variable
\(\begin{align*} I & = \int \frac{1}{csc^3x}dx \\
& = \int sin^3x.dx \\
& = \int sin²x.sinx.dx \\
& = \int (1-cos²x).sinx.dx \\ \end{align*}\)
Faisons le Changement de variable: \( t=cosx \Rightarrow dt = -sinx.dx\)
\(\begin{align*} I & = \int (1-cos²x).sinx.dx \\
& = \int (1-t²)(-dt) \\
& = \int (t²-1)dt \\
& = \frac{1}{3}t^3-t +C \\
& = \frac{1}{3}cos^3x-cosx +C
\end{align*}\)
\[\boxed {\begin{align*} I & = \frac{1}{3}cos^3x-cosx +C(\in \mathbb R) \end{align*}}\]