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. $${\large\int {\frac{{dx}}{{{{\left( {{x^2} – {a^2}} \right)}^n}}}}\normalsize} =$$ $$-{\large\frac{x}{{2\left( {n – 1} \right){a^2}{{\left( {{x^2} – {a^2}} \right)}^{n – 1}}}}\normalsize}$$ $$-\;{\large\frac{{2n – 3}}{{2\left( {n – 1} \right){a^2}}}\int\normalsize} {\large\frac{{dx}}{{{{\left( {{x^2} – {a^2}} \right)}^{n – 1}}}}\normalsize} ,$$ $$n \ne 1.$$