||V{maketoc}||
^@#16:
!جدول انتگرال معکوس توابع مثلثاتی
@@{TEX()} {\int \arctan{x} \, dx = x \, \arctan{x} - \frac{1}{2} \ln{\left| 1 + x^2\right|} + C} {TEX}@@
---
!!@@Arcsin@@
@@{TEX()} {\int \arcsin \frac{x}{c} \ dx = x \arcsin \frac{x}{c} + \sqrt{c^2 - x^2}} {TEX}@@
@@{TEX()} {\int x \arcsin \frac{x}{c} \ dx = \left( \frac{x^2}{2} - \frac{c^2}{4} \right) \arcsin \frac{x}{c} + \frac{x}{4} \sqrt{c^2 - x^2}} {TEX}@@
@@{TEX()} {\int x^2 \arcsin \frac{x}{c} \ dx = \frac{x^3}{3} \arcsin \frac{x}{c} + \frac{x^2 + 2c^2}{9} \sqrt{c^2 - x^2}} {TEX}@@
@@{TEX()} {\int x^n \arcsin x \ dx = \frac{1}{n + 1} \left( x^{n + 1} \arcsin x + \frac{x^n \sqrt{1 - x^2} - n x^{n - 1} \arcsin x}{n - 1} + n \int x^{n - 2} \arcsin x \ dx \right)} {TEX}@@
---
!!@@Arccos@@
@@{TEX()} {\int \arccos \frac{x} \ dx = x \arccos \frac{x}- \sqrt{1^2 - x^2}} {TEX}@@
@@{TEX()} {\int x \arccos \frac{x}{c} \ dx = \left( \frac{x^2}{2} - \frac{c^2}{4} \right) \arccos \frac{x}{c} - \frac{x}{4} \sqrt{c^2 - x^2}} {TEX}@@
@@{TEX()} {\int x^2 \arccos \frac{x}{c} \ dx = \frac{x^3}{3} \arccos \frac{x}{c} - \frac{x^2 + 2c^2}{9} \sqrt{c^2 - x^2}} {TEX}@@
---
!!@@Arctan@@
@@{TEX()} {\int \arctan\bigg( \frac{x}{c}\bigg) dx = x \arctan \bigg( \frac{x}{c} \bigg) - \frac{c}{2} \ln(c^2 + x^2)} {TEX}@@
@@{TEX()} {\int x \arctan\bigg( \frac{x}{c}\bigg) dx = \frac{ (c^2 + x^2) \arctan \bigg( \frac{x}{c} \bigg) - c x}{2}} {TEX}@@
@@{TEX()} {\int x^2 \arctan\bigg( \frac{x}{c}\bigg) dx = \frac{x^3}{3} \arctan \bigg(\frac{x}{c}\bigg) - \frac{c x^2}{6} + \frac{c^3}{6} \ln{c^2 + x^2}} {TEX}@@
@@{TEX()} {\int x^n \arctan \bigg( \frac{x}{c}\bigg) dx = \frac{x^{n + 1}}{n + 1} \arctan \bigg( \frac{x}{c} \bigg) - \frac{c}{n + 1} \int \frac{x^{n + 1}}{c^2 + x^2} \ dx, \quad n \neq 1} {TEX}@@
---
!!@@arccot@@
@@{TEX()} {\int \arccot \frac{x}{c} \ dx = x \arccot \frac{x}{c} + \frac{c}{2} \ln(c^2 + x^2)} {TEX}@@
@@{TEX()} {\int x \arccot \frac{x}{c} \ dx = \frac{c^2 + x^2}{2} \arccot \frac{x}{c} + \frac{c x}{2}} {TEX}@@
@@{TEX()} {\int x^2 \arccot \frac{x}{c} \ dx = \frac{x^3}{3} \arccot \frac{x}{c} + \frac{c x^2}{6} - \frac{c^3}{6} \ln(c^2 + x^2)} {TEX}@@
@@{TEX()} {\int x^n \arccot \frac{x}{c} \ dx = \frac{x^{n + 1}}{n+1} \arccot \frac{x}{c} + \frac{c}{n + 1} \int \frac{x^{n + 1}}{c^2 + x^2} \ dx, \quad n \neq 1} {TEX}@@
---
!!@@Arcsec@@
@@{TEX()} {\int \arcsec \frac{x}{c} \ dx = x \arcsec \frac{x}{c} + \frac{x}{c |x|} \ln \left| x \pm \sqrt{x^2 - 1} \right|} {TEX}@@
@@{TEX()} {\int x \arcsec x \ dx = \frac{1}{2} \left( x^2 \arcsec x - \sqrt{x^2 - 1} \right)} {TEX}@@
@@{TEX()} {\int x^n \arcsec x \ dx = \frac{1}{n + 1} \left( x^{n + 1} \arcsec x - \frac{1}{n} \left[ x^{n - 1} \sqrt{x^2 - 1} + (1 - n) \left( x^{n - 1} \arcsec x + (1 - n) \int x^{n - 2} \arcsec x \ dx \right) \right] \right)} {TEX}@@
---
!همچنین ببینید
((جدول انتگرال توابع گویا))
((جدول انتگرال توابع گنگ))
((جدول انتگرال توابع لگاریتمی))
((جدول انتگرال توابع نمایی))
((جدول انتگرال توابع مثلثاتی))
((جدول انتگرال توابع هیپربولیک))

#@^