BerandaWikiDaftar integral dari fungsi invers hiperbolikArtikel WikipediaDaftar integral dari fungsi invers hiperbolikDaftar integral tak tentu (antiderivatif) dari fungsi invers hiperbolik. Untuk daftar lengkap fungsi integral, lihat Tabel integral.Dalam semua rumus, konstanta a diasumsikan bukan nol, dan C melambangkan konstanta integrasi. Untuk setiap rumus integrasi invers hiperbolik di bawah ini ada rumus yang bersangkutan dalam daftar integral dari fungsi invers trigonometri.artikel daftar WikimediaDiperbarui 26 Januari 2017Sumber: Lihat artikel asli di WikipediaDaftar integral tak tentu (antiderivatif) dari fungsi invers hiperbolik. Untuk daftar lengkap fungsi integral, lihat Tabel integral. Dalam semua rumus, konstanta a diasumsikan bukan nol, dan C melambangkan konstanta integrasi. Untuk setiap rumus integrasi invers hiperbolik di bawah ini ada rumus yang bersangkutan dalam daftar integral dari fungsi invers trigonometri. Rumus integrasi invers hiperbolik sinus ∫ arsinh ( a x ) d x = x arsinh ( a x ) − a 2 x 2 + 1 a + C {\displaystyle \int \operatorname {arsinh} (a\,x)\,dx=x\,\operatorname {arsinh} (a\,x)-{\frac {\sqrt {a^{2}\,x^{2}+1}}{a}}+C} ∫ x arsinh ( a x ) d x = x 2 arsinh ( a x ) 2 + arsinh ( a x ) 4 a 2 − x a 2 x 2 + 1 4 a + C {\displaystyle \int x\,\operatorname {arsinh} (a\,x)dx={\frac {x^{2}\,\operatorname {arsinh} (a\,x)}{2}}+{\frac {\operatorname {arsinh} (a\,x)}{4\,a^{2}}}-{\frac {x{\sqrt {a^{2}\,x^{2}+1}}}{4\,a}}+C} ∫ x 2 arsinh ( a x ) d x = x 3 arsinh ( a x ) 3 − ( a 2 x 2 − 2 ) a 2 x 2 + 1 9 a 3 + C {\displaystyle \int x^{2}\,\operatorname {arsinh} (a\,x)dx={\frac {x^{3}\,\operatorname {arsinh} (a\,x)}{3}}-{\frac {\left(a^{2}\,x^{2}-2\right){\sqrt {a^{2}\,x^{2}+1}}}{9\,a^{3}}}+C} ∫ x m arsinh ( a x ) d x = x m + 1 arsinh ( a x ) m + 1 − a m + 1 ∫ x m + 1 a 2 x 2 + 1 d x ( m ≠ − 1 ) {\displaystyle \int x^{m}\,\operatorname {arsinh} (a\,x)dx={\frac {x^{m+1}\,\operatorname {arsinh} (a\,x)}{m+1}}\,-\,{\frac {a}{m+1}}\int {\frac {x^{m+1}}{\sqrt {a^{2}\,x^{2}+1}}}\,dx\quad (m\neq -1)} ∫ arsinh ( a x ) 2 d x = 2 x + x arsinh ( a x ) 2 − 2 a 2 x 2 + 1 arsinh ( a x ) a + C {\displaystyle \int \operatorname {arsinh} (a\,x)^{2}\,dx=2\,x+x\,\operatorname {arsinh} (a\,x)^{2}-{\frac {2\,{\sqrt {a^{2}\,x^{2}+1}}\,\operatorname {arsinh} (a\,x)}{a}}+C} ∫ arsinh ( a x ) n d x = x arsinh ( a x ) n − n a 2 x 2 + 1 arsinh ( a x ) n − 1 a + n ( n − 1 ) ∫ arsinh ( a x ) n − 2 d x {\displaystyle \int \operatorname {arsinh} (a\,x)^{n}\,dx=x\,\operatorname {arsinh} (a\,x)^{n}\,-\,{\frac {n\,{\sqrt {a^{2}\,x^{2}+1}}\,\operatorname {arsinh} (a\,x)^{n-1}}{a}}\,+\,n\,(n-1)\int \operatorname {arsinh} (a\,x)^{n-2}\,dx} ∫ arsinh ( a x ) n d x = − x arsinh ( a x ) n + 2 ( n + 1 ) ( n + 2 ) + a 2 x 2 + 1 arsinh ( a x ) n + 1 a ( n + 1 ) + 1 ( n + 1 ) ( n + 2 ) ∫ arsinh ( a x ) n + 2 d x ( n ≠ − 1 , − 2 ) {\displaystyle \int \operatorname {arsinh} (a\,x)^{n}\,dx=-{\frac {x\,\operatorname {arsinh} (a\,x)^{n+2}}{(n+1)\,(n+2)}}\,+\,{\frac {{\sqrt {a^{2}\,x^{2}+1}}\,\operatorname {arsinh} (a\,x)^{n+1}}{a(n+1)}}\,+\,{\frac {1}{(n+1)\,(n+2)}}\int \operatorname {arsinh} (a\,x)^{n+2}\,dx\quad (n\neq -1,-2)} Rumus integrasi invers hiperbolik kosinus ∫ arcosh ( a x ) d x = x arcosh ( a x ) − a x + 1 a x − 1 a + C {\displaystyle \int \operatorname {arcosh} (a\,x)\,dx=x\,\operatorname {arcosh} (a\,x)-{\frac {{\sqrt {a\,x+1}}\,{\sqrt {a\,x-1}}}{a}}+C} ∫ x arcosh ( a x ) d x = x 2 arcosh ( a x ) 2 − arcosh ( a x ) 4 a 2 − x a x + 1 a x − 1 4 a + C {\displaystyle \int x\,\operatorname {arcosh} (a\,x)dx={\frac {x^{2}\,\operatorname {arcosh} (a\,x)}{2}}-{\frac {\operatorname {arcosh} (a\,x)}{4\,a^{2}}}-{\frac {x\,{\sqrt {a\,x+1}}\,{\sqrt {a\,x-1}}}{4\,a}}+C} ∫ x 2 arcosh ( a x ) d x = x 3 arcosh ( a x ) 3 − ( a 2 x 2 + 2 ) a x + 1 a x − 1 9 a 3 + C {\displaystyle \int x^{2}\,\operatorname {arcosh} (a\,x)dx={\frac {x^{3}\,\operatorname {arcosh} (a\,x)}{3}}-{\frac {\left(a^{2}\,x^{2}+2\right){\sqrt {a\,x+1}}\,{\sqrt {a\,x-1}}}{9\,a^{3}}}+C} ∫ x m arcosh ( a x ) d x = x m + 1 arcosh ( a x ) m + 1 − a m + 1 ∫ x m + 1 a x + 1 a x − 1 d x ( m ≠ − 1 ) {\displaystyle \int x^{m}\,\operatorname {arcosh} (a\,x)dx={\frac {x^{m+1}\,\operatorname {arcosh} (a\,x)}{m+1}}\,-\,{\frac {a}{m+1}}\int {\frac {x^{m+1}}{{\sqrt {a\,x+1}}\,{\sqrt {a\,x-1}}}}\,dx\quad (m\neq -1)} ∫ arcosh ( a x ) 2 d x = 2 x + x arcosh ( a x ) 2 − 2 a x + 1 a x − 1 arcosh ( a x ) a + C {\displaystyle \int \operatorname {arcosh} (a\,x)^{2}\,dx=2\,x+x\,\operatorname {arcosh} (a\,x)^{2}-{\frac {2\,{\sqrt {a\,x+1}}\,{\sqrt {a\,x-1}}\,\operatorname {arcosh} (a\,x)}{a}}+C} ∫ arcosh ( a x ) n d x = x arcosh ( a x ) n − n a x + 1 a x − 1 arcosh ( a x ) n − 1 a + n ( n − 1 ) ∫ arcosh ( a x ) n − 2 d x {\displaystyle \int \operatorname {arcosh} (a\,x)^{n}\,dx=x\,\operatorname {arcosh} (a\,x)^{n}\,-\,{\frac {n\,{\sqrt {a\,x+1}}\,{\sqrt {a\,x-1}}\,\operatorname {arcosh} (a\,x)^{n-1}}{a}}\,+\,n\,(n-1)\int \operatorname {arcosh} (a\,x)^{n-2}\,dx} ∫ arcosh ( a x ) n d x = − x arcosh ( a x ) n + 2 ( n + 1 ) ( n + 2 ) + a x + 1 a x − 1 arcosh ( a x ) n + 1 a ( n + 1 ) + 1 ( n + 1 ) ( n + 2 ) ∫ arcosh ( a x ) n + 2 d x ( n ≠ − 1 , − 2 ) {\displaystyle \int \operatorname {arcosh} (a\,x)^{n}\,dx=-{\frac {x\,\operatorname {arcosh} (a\,x)^{n+2}}{(n+1)\,(n+2)}}\,+\,{\frac {{\sqrt {a\,x+1}}\,{\sqrt {a\,x-1}}\,\operatorname {arcosh} (a\,x)^{n+1}}{a\,(n+1)}}\,+\,{\frac {1}{(n+1)\,(n+2)}}\int \operatorname {arcosh} (a\,x)^{n+2}\,dx\quad (n\neq -1,-2)} Rumus integrasi invers hiperbolik tangen ∫ artanh ( a x ) d x = x artanh ( a x ) + ln ( 1 − a 2 x 2 ) 2 a + C {\displaystyle \int \operatorname {artanh} (a\,x)\,dx=x\,\operatorname {artanh} (a\,x)+{\frac {\ln \left(1-a^{2}\,x^{2}\right)}{2\,a}}+C} ∫ x artanh ( a x ) d x = x 2 artanh ( a x ) 2 − artanh ( a x ) 2 a 2 + x 2 a + C {\displaystyle \int x\,\operatorname {artanh} (a\,x)dx={\frac {x^{2}\,\operatorname {artanh} (a\,x)}{2}}-{\frac {\operatorname {artanh} (a\,x)}{2\,a^{2}}}+{\frac {x}{2\,a}}+C} ∫ x 2 artanh ( a x ) d x = x 3 artanh ( a x ) 3 + ln ( 1 − a 2 x 2 ) 6 a 3 + x 2 6 a + C {\displaystyle \int x^{2}\,\operatorname {artanh} (a\,x)dx={\frac {x^{3}\,\operatorname {artanh} (a\,x)}{3}}+{\frac {\ln \left(1-a^{2}\,x^{2}\right)}{6\,a^{3}}}+{\frac {x^{2}}{6\,a}}+C} ∫ x m artanh ( a x ) d x = x m + 1 artanh ( a x ) m + 1 − a m + 1 ∫ x m + 1 1 − a 2 x 2 d x ( m ≠ − 1 ) {\displaystyle \int x^{m}\,\operatorname {artanh} (a\,x)dx={\frac {x^{m+1}\operatorname {artanh} (a\,x)}{m+1}}-{\frac {a}{m+1}}\int {\frac {x^{m+1}}{1-a^{2}\,x^{2}}}\,dx\quad (m\neq -1)} Rumus integrasi invers hiperbolik kotangen ∫ arcoth ( a x ) d x = x arcoth ( a x ) + ln ( a 2 x 2 − 1 ) 2 a + C {\displaystyle \int \operatorname {arcoth} (a\,x)\,dx=x\,\operatorname {arcoth} (a\,x)+{\frac {\ln \left(a^{2}\,x^{2}-1\right)}{2\,a}}+C} ∫ x arcoth ( a x ) d x = x 2 arcoth ( a x ) 2 − arcoth ( a x ) 2 a 2 + x 2 a + C {\displaystyle \int x\,\operatorname {arcoth} (a\,x)dx={\frac {x^{2}\,\operatorname {arcoth} (a\,x)}{2}}-{\frac {\operatorname {arcoth} (a\,x)}{2\,a^{2}}}+{\frac {x}{2\,a}}+C} ∫ x 2 arcoth ( a x ) d x = x 3 arcoth ( a x ) 3 + ln ( a 2 x 2 − 1 ) 6 a 3 + x 2 6 a + C {\displaystyle \int x^{2}\,\operatorname {arcoth} (a\,x)dx={\frac {x^{3}\,\operatorname {arcoth} (a\,x)}{3}}+{\frac {\ln \left(a^{2}\,x^{2}-1\right)}{6\,a^{3}}}+{\frac {x^{2}}{6\,a}}+C} ∫ x m arcoth ( a x ) d x = x m + 1 arcoth ( a x ) m + 1 + a m + 1 ∫ x m + 1 a 2 x 2 − 1 d x ( m ≠ − 1 ) {\displaystyle \int x^{m}\,\operatorname {arcoth} (a\,x)dx={\frac {x^{m+1}\operatorname {arcoth} (a\,x)}{m+1}}+{\frac {a}{m+1}}\int {\frac {x^{m+1}}{a^{2}\,x^{2}-1}}\,dx\quad (m\neq -1)} Rumus integrasi invers hiperbolik sekan ∫ arsech ( a x ) d x = x arsech ( a x ) − 2 a arctan 1 − a x 1 + a x + C {\displaystyle \int \operatorname {arsech} (a\,x)\,dx=x\,\operatorname {arsech} (a\,x)-{\frac {2}{a}}\,\operatorname {arctan} {\sqrt {\frac {1-a\,x}{1+a\,x}}}+C} ∫ x arsech ( a x ) d x = x 2 arsech ( a x ) 2 − ( 1 + a x ) 2 a 2 1 − a x 1 + a x + C {\displaystyle \int x\,\operatorname {arsech} (a\,x)dx={\frac {x^{2}\,\operatorname {arsech} (a\,x)}{2}}-{\frac {(1+a\,x)}{2\,a^{2}}}{\sqrt {\frac {1-a\,x}{1+a\,x}}}+C} ∫ x 2 arsech ( a x ) d x = x 3 arsech ( a x ) 3 − 1 3 a 3 arctan 1 − a x 1 + a x − x ( 1 + a x ) 6 a 2 1 − a x 1 + a x + C {\displaystyle \int x^{2}\,\operatorname {arsech} (a\,x)dx={\frac {x^{3}\,\operatorname {arsech} (a\,x)}{3}}\,-\,{\frac {1}{3\,a^{3}}}\,\operatorname {arctan} {\sqrt {\frac {1-a\,x}{1+a\,x}}}\,-\,{\frac {x(1+a\,x)}{6\,a^{2}}}{\sqrt {\frac {1-a\,x}{1+a\,x}}}\,+\,C} ∫ x m arsech ( a x ) d x = x m + 1 arsech ( a x ) m + 1 + 1 m + 1 ∫ x m ( 1 + a x ) 1 − a x 1 + a x d x ( m ≠ − 1 ) {\displaystyle \int x^{m}\,\operatorname {arsech} (a\,x)dx={\frac {x^{m+1}\,\operatorname {arsech} (a\,x)}{m+1}}\,+\,{\frac {1}{m+1}}\int {\frac {x^{m}}{(1+a\,x){\sqrt {\frac {1-a\,x}{1+a\,x}}}}}\,dx\quad (m\neq -1)} Rumus integrasi invers hiperbolik kosekan ∫ arcsch ( a x ) d x = x arcsch ( a x ) + 1 a arcoth 1 a 2 x 2 + 1 + C {\displaystyle \int \operatorname {arcsch} (a\,x)\,dx=x\,\operatorname {arcsch} (a\,x)+{\frac {1}{a}}\,\operatorname {arcoth} {\sqrt {{\frac {1}{a^{2}\,x^{2}}}+1}}+C} ∫ x arcsch ( a x ) d x = x 2 arcsch ( a x ) 2 + x 2 a 1 a 2 x 2 + 1 + C {\displaystyle \int x\,\operatorname {arcsch} (a\,x)dx={\frac {x^{2}\,\operatorname {arcsch} (a\,x)}{2}}+{\frac {x}{2\,a}}{\sqrt {{\frac {1}{a^{2}\,x^{2}}}+1}}+C} ∫ x 2 arcsch ( a x ) d x = x 3 arcsch ( a x ) 3 − 1 6 a 3 arcoth 1 a 2 x 2 + 1 + x 2 6 a 1 a 2 x 2 + 1 + C {\displaystyle \int x^{2}\,\operatorname {arcsch} (a\,x)dx={\frac {x^{3}\,\operatorname {arcsch} (a\,x)}{3}}\,-\,{\frac {1}{6\,a^{3}}}\,\operatorname {arcoth} {\sqrt {{\frac {1}{a^{2}\,x^{2}}}+1}}\,+\,{\frac {x^{2}}{6\,a}}{\sqrt {{\frac {1}{a^{2}\,x^{2}}}+1}}\,+\,C} ∫ x m arcsch ( a x ) d x = x m + 1 arcsch ( a x ) m + 1 + 1 a ( m + 1 ) ∫ x m − 1 1 a 2 x 2 + 1 d x ( m ≠ − 1 ) {\displaystyle \int x^{m}\,\operatorname {arcsch} (a\,x)dx={\frac {x^{m+1}\operatorname {arcsch} (a\,x)}{m+1}}\,+\,{\frac {1}{a(m+1)}}\int {\frac {x^{m-1}}{\sqrt {{\frac {1}{a^{2}\,x^{2}}}+1}}}\,dx\quad (m\neq -1)} lbsDaftar integralFungsi rasional • Fungsi irrasional • Fungsi trigonometri • Invers trigonometri • Fungsi hiperbolik • Invers hiperbolik • Fungsi eksponensial • Fungsi logaritmik Bagikan artikel iniShare:
