# Reduction Formulas of Integration

1) $\int {{{\sin }^n}xdx = – \frac{{\cos x{{\sin }^{n – 1}}x}}{n} + \frac{{n – 1}}{n}\int {{{\sin }^{n – 2}}xdx} }$

2) $\int {{{\cos }^n}xdx = \frac{{\sin x{{\cos }^{n – 1}}x}}{n} + \frac{{n – 1}}{n}\int {{{\cos }^{n – 2}}xdx} }$

3) $\int {{{\tan }^n}xdx = \frac{{{{\tan }^{n – 1}}x}}{{n – 1}} – \int {{{\tan }^{n – 2}}xdx} }$

4) $\int {{{\cot }^n}xdx = – \frac{{{{\cot }^{n – 1}}x}}{{n – 1}} – \int {{{\cot }^{n – 2}}xdx} }$

5) $\int {{{\sec }^n}xdx = \frac{{\tan x{{\sec }^{n – 2}}x}}{{n – 1}}} + \frac{{n – 2}}{{n – 1}}\int {{{\sec }^{n – 2}}xdx}$

6) $\int {{{\csc }^n}xdx = – \frac{{\cot x{{\csc }^{n – 2}}x}}{{n – 1}}} + \frac{{n – 2}}{{n – 1}}\int {{{\csc }^{n – 2}}xdx}$

7) $\int {{{\sin }^m}x{{\cos }^n}xdx = – \frac{1}{{m + n}}{{\sin }^{m – 1}}x{{\cos }^{n + 1}}x + \frac{{m – 1}}{{m + n}}\int {{{\sin }^{m – 2}}x{{\cos }^n}xdx} }$

8) $\int {\frac{{{{\cos }^n}x}}{{{{\sin }^m}x}}dx = – \frac{{{{\cos }^{n + 1}}x}}{{(m – 1){{\sin }^{m – 1}}x}} – \frac{{n – m + 2}}{{m – 1}}\int {\frac{{{{\cos }^n}x}}{{{{\sin }^{m – 2}}x}}dx} }$

9) $\int {\frac{{{{\sin }^m}x}}{{{{\cos }^n}x}}dx = – \frac{{{{\sin }^{m + 1}}x}}{{(n – 1){{\cos }^{n – 1}}x}} – \frac{{m – n + 2}}{{n – 1}}\int {\frac{{{{\sin }^m}x}}{{{{\cos }^{n – 2}}x}}dx} }$

10) $\int {{{\cos }^m}x\cos nxdx = \frac{{{{\cos }^m}x\sin nx}}{{m + n}} + \frac{m}{{m + n}}\int {{{\cos }^{m – 1}}x\cos (n – 1)xdx} }$

11) $\int {{{\sin }^m}x\sin nxdx = \frac{{n{{\sin }^m}x\cos nx}}{{{m^2} – {n^2}}} – \frac{m}{{{m^2} – {n^2}}}{{\sin }^{m – 1}}x\cos x\sin nx + \frac{{m(m – 1)}}{{{m^2} – {n^2}}}\int {{{\sin }^{m – 2}}x\sin nxdx} }$

12) $\int {{{\cos }^m}x\sin nxdx = \frac{1}{{m + n}}( – {{\cos }^m}x\cos nx) + \frac{m}{{m + n}}\int {{{\cos }^{m – 1}}x\sin (n – 1)dx} }$

13) $\int {{{\sin }^m}x\cos nxdx = \frac{{m\cos x\cos nx + n\sin x\sin nx}}{{{n^2} – {m^2}}}{{\sin }^{m – 1}}x – \frac{{m(m – 1)}}{{{n^2} – {m^2}}}\int {{{\sin }^{m – 2}}x\cos nxdx} }$

14) $\int {{x^n}{e^{ax}}dx = \frac{{{x^n}{e^{ax}}}}{a} – \frac{n}{a}\int {{x^{n – 1}}{e^{ax}}dx} }$

15) $\int {{x^m}\sin nxdx = – \frac{{{x^m}\cos nx}}{n} + \frac{m}{{{n^2}}}{x^{m – 1}}\sin nx – \frac{{m(m – 1)}}{{{n^2}}}\int {{x^{n – 2}}\sin nxdx} }$

16) $\int {{x^m}\sin nxdx = – \frac{{{x^m}\sin nx}}{n} + \frac{m}{{{n^2}}}{x^{m – 1}}\cos nx – \frac{{m(m – 1)}}{{{n^2}}}\int {{x^{m – 2}}\cos nxdx} }$

17) $\int\limits_0^{\frac{\pi }{2}} {{{\sin }^n}xdx = } \left\{ {\begin{array}{*{20}{c}}\\ {\frac{{(n – 1)(n – 3)(n – 5) \cdots 2}}{{n(n – 2)(n – 4) \cdots 3}},{\text{ if }}n{\text{ is odd}}} \\ {\frac{{(n – 1)(n – 3)(n – 5) \cdots 2}}{{n(n – 2)(n – 4) \cdots 2}} \cdot \frac{\pi }{2},{\text{ if }}n{\text{ is even}}} \\ \end{array}} \right.$

18) $\int\limits_0^{\frac{\pi }{2}} {{{\cos }^n}xdx = } \left\{ {\begin{array}{*{20}{c}} {\frac{{(n – 1)(n – 3)(n – 5) \cdots 2}}{{n(n – 2)(n – 4) \cdots 3}},{\text{ if }}n{\text{ is odd}}} \\ {\frac{{(n – 1)(n – 3)(n – 5) \cdots 2}}{{n(n – 2)(n – 4) \cdots 2}} \cdot \frac{\pi }{2},{\text{ if }}n{\text{ is even}}} \\ \end{array}} \right.$

The formulas (17) and (18) are called Wallis Formulas.