Kenan kılıçaslan

Integrali contenenti \(\tan(a x)\)

Questa pagina presenta le formule degli integrali indefiniti contenenti funzioni trigonometriche della forma \(\tan(a x)\). Questi integrali si risolvono generalmente mediante trasformazioni trigonometriche e espressioni logaritmiche.

1.
\(\small \displaystyle\int\tan ax dx=-\displaystyle \frac{1}{a}\ln\cos ax=\displaystyle \frac{1}{a}\ln\sec ax\)
2.
\(\small \displaystyle\int\tan^2 ax dx=\displaystyle \frac{\tan ax}{a}-x\)
3.
\(\small \displaystyle\int\tan^3 ax dx=\displaystyle \frac{\tan^2 ax}{2a}+\displaystyle \frac{1}{a}\ln\cos ax\)
4.
\(\small \displaystyle\int\tan^n ax \sec^2 ax dx=\displaystyle \frac{\tan^{n+1}ax}{(n+1)a}\)
5.
\(\small \displaystyle\int\displaystyle \frac{\sec^2 ax}{\tan ax}dx=\displaystyle \frac{1}{a}\ln\tan ax\)
6.
\(\small \displaystyle\int\displaystyle \frac{dx}{\tan ax}=\displaystyle \frac{1}{a}\ln\sin ax\)
7.
\(\small \displaystyle\int x\tan^2 ax dx=\displaystyle \frac{x\tan ax}{a}+\displaystyle \frac{1}{a^2}\ln\cos ax-\displaystyle \frac{x^2}{2}\)
8.
\(\small \displaystyle\int\displaystyle \frac{dx}{p+q\tan ax}=\displaystyle \frac{px}{p^2+q^2}+\displaystyle \frac{q}{a(p^2+q^2)}\ln(q\sin ax+p\cos ax)\)
9.
\(\small \displaystyle\int\tan^n ax dx=\displaystyle \frac{\tan^{n-1}ax}{(n-1)a}-\int\tan^{n-2}ax dx\)