Reduction Formulas

There is no universal technique known, which will solve any indefinite integral. There are certain techniques, such as substitution, partial fractions, integration by parts; but these apply to only certain types of integrals.

A reduction formula is another such technique in the form of recurrence relation.

  • Reduction Formulas

A complicated integral is expressed using simpler integrals of the same kind. The integrals involve an integer parameter. The complicated integral I_n is represented in terms of I_{n-1} or I_{n-2}, where n is an integer. Ultimately, when n=1 or n=2, I_{1} or I_2 can be evaluated easily.

So, we reduce I_n to a simpler form to get the answer.

  • Reduction Formulas for Trigonometric Functions

2 cases arise for such integrals, where only 1 integer parameter (n) is there; n can be odd or even.

{\int \limits_0^{\frac {\pi} 2}cos^nx \ dx = \int \limits_0^{\frac {\pi} 2}sin^nx \ dx = \frac {(n-1) \times .... subtract \ 2 ..... 2 \ or \ 1}{(n) \times .... subtract \ 2 ..... 2 \ or \ 1}}

{{\times \frac {\pi}{2}, if \ n \ is \ even}}

{\int tan^n x dx = \frac {tan^{n-1}x}{n-1} - \int tan^{n-2}xdx}

When 2 integer parameters are there, m and n, we have 2 cases; when both m and n are even and otherwise.

{\int \limits_{0}^{\frac {\pi}{2}} sin^mx cos^n x dx = \Big \{\frac {[(m-1) \times .... subtract \ 2 ..... 2 \ or \ 1] \times [(n-1) \times .... subtract \ 2 ..... 2 \ or \ 1]}{[(m+n)\times (m+n-2) \times ..... subtract \ 2 \ ... 2.... 2 \ or \ 1 ]} \Big \}}

{{\times \frac {\pi}{2}, if \ m,\ n \ are \ even}}

{\times 1, \ otherwise}

  • Reduction Formulas for Product of Algebraic and Trigonometric Functions

In these problems, we will have functions of the kind x,x^2 etc. and trigonometric functions. We will use integration by parts and keep the first function algebraic. (i.e. u).

  • Reduction Formulas for Product of Exponential and Trigonometric Functions

In these problems, we will have functions of the kind e^{x},e^{-x} etc. and trigonometric functions. We will use integration by parts and keep the first function trigonometric. (i.e. u).

  • Reduction Formulas for Product of Algebraic and Other Functions

In this case, we have to choose the first function u to get the answer by integration by parts. There is no rule of thumb for the choice of first function. It totally depends on the functions in the integrand.

  • Useful Information for Definite Integrals

A definite integral has lower and upper limits. For example,

{I = \int \limits_a^b f(x)dx}

If the limits are such that a = -b, check whether f(x) is even or odd. If it is odd, then I is 0. If it is even, then

{I = 2 \times \int \limits_0^b f(x)dx}

Also, for any definite integral, following relation is true:

{\int \limits_a^b f(x)dx = \int \limits_a^b f(a+b-x)dx}

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