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} } <br />

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.