# 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:

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