Formulas

Calculus

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Reduction Formulas for Integrals

  • Functions: \({e^x},\) \({x^n},\) \(\sinh x,\) \(\cosh x,\) \(\tanh x,\) \(\text {coth }x,\) \(\text {sech }x,\) \(\text {csch }x,\) \(\sin x,\) \(\cos x,\) \(\tan x,\) \(\cot x,\) \(\ln x,\) \(\arcsin x,\) \(\arccos x,\) \(\arctan x\)
    Argument (independent variable): \(x\)
    Natural numbers: \(n\), \(m\)
    Real numbers: \(a\), \(b\), \(c\)
    1. To find some integrals we can use the reduction formulas. These formulas enable us to reduce the degree of the integrand and calculate the integrals in a finite number of steps. Below are the reduction formulas for integrals involving the most common functions.
    2. \({\large\int\normalsize} {{x^n}{e^{mx}}dx} =\) \( {\large\frac{1}{m}\normalsize}{x^n}{e^{mx}} \) \(-\; {\large\frac{n}{m}\normalsize} {\large\int\normalsize} {{x^{n – 1}}{e^{mx}}dx} \)
    3. \({\large\int\normalsize} {{\large\frac{{{e^{mx}}}}{{{x^n}}}\normalsize} dx} =\) \( – {\large\frac{{{e^{mx}}}}{{\left( {n – 1} \right){x^{n – 1}}}}\normalsize} \) \(+\; {\large\frac{m}{{n – 1}}\normalsize} {\large\int\normalsize} {{\large\frac{{{e^{mx}}}}{{{x^{n – 1}}}}\normalsize} dx} ,\) \(n \ne 1.\)
    4. \({\large\int\normalsize} {{{\sinh }^n}x\,dx} =\) \( – {\large\frac{1}{n}\normalsize}{\sinh ^{n – 1}}x\cosh x \) \(-\; {\large\frac{{n – 1}}{n}\normalsize} {\large\int\normalsize} {{{\sinh }^{n – 2}}x\,dx} \)
    5. \({\large\int\normalsize} {\large\frac{{dx}}{{{{\sinh }^n}x}}\normalsize} =\) \( – {\large\frac{{\cosh x}}{{\left( {n – 1} \right){{\sinh }^{n – 1}}x}}\normalsize} \) \(+\;{\large\frac{{n – 2}}{{n – 1}}\int\normalsize} {\large\frac{{dx}}{{{{\sinh }^{n – 2}}x}}\normalsize} ,\) \(n \ne 1.\)
    6. \({\large\int\normalsize} {{\cosh^n}x\,dx} =\) \({\large\frac{1}{n}\normalsize}\sinh x\,{\cosh^{n – 1}}x \) \(-\;{\large\frac{{n – 1}}{n}\int\normalsize} {{\cosh^{n – 2}}x\,dx} \)
    7. \({\large\int\normalsize} {\large\frac{{dx}}{{{\cosh^n}x}}\normalsize} =\) \( – {\large\frac{{\sinh x}}{{\left( {n – 1} \right){\cosh^{n – 1}}x}}\normalsize} \) \(+\;{\large\frac{{n – 2}}{{n – 1}}\int\normalsize} {\large\frac{{dx}}{{{\cosh^{n – 2}}x}}\normalsize} ,\) \(n \ne 1.\)
    8. \({\large\int\normalsize} {{\sinh^n}x\,{\cosh^m}x\,dx} =\) \({\large\frac{{{\sinh^{n + 1}}x\,{\cosh^{m – 1}}x}}{{n + m}}\normalsize} \) \(+\;{\large\frac{{m – 1}}{{n + m}}\int\normalsize} {{\sinh^n}x\,{\cosh^{m – 2}}x\,dx} \)
    9. \({\large\int\normalsize} {{\sinh^n}x\,{\cosh^m}x\,dx} =\) \({\large\frac{{{\sinh^{n – 1}}x\,{\cosh^{m + 1}}x}}{{n + m}}\normalsize} \) \(-\; {\large\frac{{n – 1}}{{n + m}}\int\normalsize} {{\sinh^{n – 2}}x\,{\cosh^m}x\,dx} \)
    10. \({\large\int\normalsize} {{\tanh^n}x\,dx} =\) \( – {\large\frac{1}{{n – 1}}\normalsize} {\tanh^{n – 1}}x \) \(+\; {\large\int\normalsize} {{\tanh^{n – 2}}x\,dx} ,\) \(n \ne 1.\)
    11. \({\large\int\normalsize} {{\coth^n}x\,dx} =\) \( – {\large\frac{1}{{n – 1}}\normalsize} {\coth^{n – 1}}x \) \(+\;{\large\int\normalsize} {{\coth^{n – 2}}x\,dx} ,\) \(n \ne 1.\)
    12. \({\large\int\normalsize} {{\text{sech}^n}x\,dx} =\) \({\large\frac{{{\text{sech}^{n – 2}}x\tanh x}}{{n – 1}}\normalsize} \) \(+\;{\large\frac{{n – 2}}{{n – 1}}\int\normalsize} {{\text{sech}^{n – 2}}x\,dx} ,\) \(n \ne 1.\)
    13. \({\large\int\normalsize} {{\sin^n}x\,dx} =\) \( -{\large\frac{1}{n}\normalsize}{\sin ^{n – 1}}x\cos x \) \(+\;{\large\frac{{n – 1}}{n}\int\normalsize} {{\sin^{n – 2}}x\,dx} \)
    14. \({\large\int\normalsize} {\large\frac{{dx}}{{{\sin^n}x}}\normalsize} =\) \( -{\large\frac{{\cos x}}{{\left( {n – 1} \right){{\sin }^{n – 1}}x}\normalsize}} \) \(+\;{\large\frac{{n – 2}}{{n – 1}}\int\normalsize} {\large\frac{{dx}}{{{\sin^{n – 2}}x}}\normalsize} ,\) \(n \ne 1.\)
    15. \({\large\int\normalsize} {{\cos^n}x\,dx} =\) \({\large\frac{1}{n}\normalsize} \sin x\,{\cos^{n – 1}}x \) \(+\;{ \large\frac{{n – 1}}{n}\int\normalsize} {{\cos^{n – 2}}x\,dx} \)
    16. \({\large\int\normalsize} {\large\frac{{dx}}{{{\cos^n}x}}\normalsize} =\) \( {\large\frac{{\sin x}}{{\left( {n – 1} \right){{\cos }^{n – 1}}x}\normalsize}} \) \(+\;{\large\frac{{n – 2}}{{n – 1}}\int\normalsize} {\large\frac{{dx}}{{{\cos^{n – 2}}x}}\normalsize} ,\) \(n \ne 1.\)
    17. \({\large\int\normalsize} {{\sin^n}x\,{\cos^m}x\,dx} =\) \({\large\frac{{{{\sin }^{n + 1}}x\,{\cos^{m – 1}}x}}{{n + m}}\normalsize} \) \(+\;{\large\frac{{m – 1}}{{n + m}}\int\normalsize} {{\sin^n}x\,{\cos^{m – 2}}x\,dx} \)
    18. \({\large\int\normalsize} {{\sin^n}x\,{\cos^m}x\,dx} =\) \(-{\large\frac{{{{\sin }^{n – 1}}x\,{\cos^{m + 1}}x}}{{n + m}}\normalsize} \) \(+\;{\large\frac{{n – 1}}{{n + m}}\int\normalsize} {{\sin^{n – 2}}x\,{\cos^m}x\,dx} \)
    19. \({\large\int\normalsize} {{\tan^n}x\,dx} =\) \({\large\frac{1}{{n – 1}}\normalsize} {\tan^{n – 1}}x \) \(-\;{\large\int\normalsize} {{\tan^{n – 2}}x\,dx} ,\) \(n \ne 1.\)
    20. \({\large\int\normalsize} {{\cot^n}x\,dx} =\) \(-{\large\frac{1}{{n – 1}}\normalsize} {\cot^{n – 1}}x \) \(-\;{\large\int\normalsize} {{\cot^{n – 2}}x\,dx} ,\) \(n \ne 1.\)
    21. \({\large\int\normalsize} {{\sec^n}x\,dx} =\) \({\large\frac{{{\sec^{n – 2}}x\tan x}}{{n – 1}}\normalsize} \) \(+\;{\large\frac{{n – 2}}{{n – 1}}\int\normalsize} {{\sec^{n – 2}}x\,dx} ,\) \(n \ne 1.\)
    22. \({\large\int\normalsize} {{\csc^n}x\,dx} =\) \(-{\large\frac{{{\csc^{n – 2}}x\cot x}}{{n – 1}}\normalsize} \) \(+\;{\large\frac{{n – 2}}{{n – 1}}\int\normalsize} {{\csc^{n – 2}}x\,dx} ,\) \(n \ne 1.\)
    23. \({\large\int\normalsize} {{x^n}{\ln^m}x\,dx} =\) \({\large\frac{{{x^{n + 1}}{\ln^m}x}}{{n + 1}}\normalsize} \) \(-\;{\large\frac{m}{{n + 1}}\int\normalsize} {{x^n}{\ln^{m – 1}}x\,dx} \)
    24. \({\large\int\normalsize} {\large{\frac{{{\ln^m}x}}{{{x^n}}}\normalsize}} \,dx =\) \( -{\large\frac{{{\ln^m}x}}{{\left( {n – 1} \right){x^{n + 1}}}}\normalsize} \) \(+\;{\large\frac{m}{{n – 1}}\int\normalsize} {{\large\frac{{{\ln^{m – 1}}x}}{{{x^n}}}\normalsize} dx},\) \(n \ne 1.\)
    25. \({\large\int\normalsize} {{\ln^n}x\,dx} =\) \(x\,{\ln ^n}x – n{\large\int\normalsize} {{\ln^{n – 1}}x\,dx} \)
    26. \({\large\int\normalsize} {{x^n}\sinh x\,dx} =\) \({x^n}\cosh x \) \(-\; n{\large\int\normalsize} {{x^{n – 1}}\cosh x\,dx} \)
    27. \({\large\int\normalsize} {{x^n}\cosh x\,dx} =\) \({x^n}\sinh x \) \(-\; n{\large\int\normalsize} {{x^{n – 1}}\sinh x\,dx} \)
    28. \({\large\int\normalsize} {{x^n}\sin x\,dx} =\) \( -{x^n}\cos x \) \(+\; n{\large\int\normalsize} {{x^{n – 1}}\cos x\,dx} \)
    29. \({\large\int\normalsize} {{x^n}\cos x\,dx} =\) \({x^n}\sin x \) \(-\; n{\large\int\normalsize} {{x^{n – 1}}\sin x\,dx} \)
    30. \({\large\int\normalsize} {{x^n}\arcsin x\,dx} =\) \({\large\frac{{{x^{n + 1}}}}{{n + 1}}\normalsize} \arcsin x \) \(-\;{\large\frac{1}{{n + 1}}\int\normalsize} {{\large\frac{{{x^{n + 1}}}}{{\sqrt {1 – {x^2}} }}\normalsize} dx} \)
    31. \({\large\int\normalsize} {{x^n}\arccos x\,dx} =\) \({\large\frac{{{x^{n + 1}}}}{{n + 1}}\normalsize} \arccos x \) \(+\;{\large\frac{1}{{n + 1}}\int\normalsize} {{\large\frac{{{x^{n + 1}}}}{{\sqrt {1 – {x^2}} }}\normalsize} dx} \)
    32. \({\large\int\normalsize} {{x^n}\arctan x\,dx} =\) \({\large\frac{{{x^{n + 1}}}}{{n + 1}}\normalsize} \arctan x \) \(-\;{\large\frac{1}{{n + 1}}\int\normalsize} {{\large\frac{{{x^{n + 1}}}}{{ 1 + {x^2} }}\normalsize} dx} \)
    33. \({\large\int\normalsize} {\large{\frac{{{x^n}}}{{a{x^n} + b}}\normalsize}} \,dx =\) \({\large\frac{x}{a}\normalsize} – {\large\frac{b}{a}\int {\frac{{dx}}{{a{x^n} + b}}}\normalsize} \)
    34. \({\large\int {\frac{{dx}}{{{{\left( {a{x^2} + bx + c} \right)}^{\,n}}}}}\normalsize} =\) \({\large\frac{{ – 2ax – b}}{{\left( {n – 1} \right)\left( {{b^2} – 4ac} \right){{\left( {a{x^2} + bx + c} \right)}^{\,n – 1}}}}\normalsize} \) \(-\;{\large\frac{{2\left( {2n – 3} \right)a}}{{\left( {n – 1} \right)\left( {{b^2} – 4ac} \right)}}\int\normalsize} {\large\frac{{dx}}{{{{\left( {a{x^2} + bx + c} \right)}^{\,n – 1}}}}\normalsize},\) \(n \ne 1.\)
    35. \({\large\int {\frac{{dx}}{{{{\left( {{x^2} + {a^2}} \right)}^n}}}}\normalsize} =\) \({\large\frac{x}{{2\left( {n – 1} \right){a^2}{{\left( {{x^2} + {a^2}} \right)}^{n – 1}}}}\normalsize} \) \(+\;{\large\frac{{2n – 3}}{{2\left( {n – 1} \right){a^2}}}\int\normalsize} {\large\frac{{dx}}{{{{\left( {{x^2} + {a^2}} \right)}^{n – 1}}}}\normalsize} ,\) \(n \ne 1.\)
    36. \({\large\int {\frac{{dx}}{{{{\left( {{x^2} – {a^2}} \right)}^n}}}}\normalsize} =\) \(-{\large\frac{x}{{2\left( {n – 1} \right){a^2}{{\left( {{x^2} – {a^2}} \right)}^{n – 1}}}}\normalsize} \) \(-\;{\large\frac{{2n – 3}}{{2\left( {n – 1} \right){a^2}}}\int\normalsize} {\large\frac{{dx}}{{{{\left( {{x^2} – {a^2}} \right)}^{n – 1}}}}\normalsize} ,\) \(n \ne 1.\)