Methods of Integration and Trigonometric Substitution
Explore methods of integration including integration by parts and trigonometric substitution. Learn how to apply these techniques to solve integrals involving logarithmic, trigonometric, and rational functions. Discover step-by-step processes and identities to simplify and evaluate various types of integrals effectively.
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Methods Methods of Integration of Integration
Steps of integration by parts: 1. Split the integration into two parts, one of which we denoted it u and the other is dv. 2. Differentiate u and integrate dv. Apply the formula: . Steps of integration by parts: 3 = dv . v . du . u u v Note once on the same question. Note: This method can be repeated more than
We use integration by parts method in the next four cases: A1. Integration of logarithmic functions when the derivative not available A2. Integration of Inverse Trigonometric functions when the derivative not available A3. Integration of two multiplied functions, one of them not derivative to the other A4. Integration of some trigonometric functions raised to a certain power
In mathematics, trigonometric substitution is replacing term or mathematical expression with a trigonometric function by using trigonometric identities, usually to simplify integrals containing radical expressions.
B1. If the integral contains the expression a a2 2 x x2 2, suppose x= cos B2. If the integral contains the expression a a2 2+ suppose x= sec B3. If the integral contains the expression x x2 suppose x= tan x= a sin z x=1 1- - sin a sin z , then use the identity sin2 2x x cos2 2x= + x x2 2, x= a tan z x =tan a tan z , then use the identity tan2 2x + sec2 2x = x +1 1 2 - - a a2 2, x= a sec z x= sec a sec z , then use the identity sec2 2x tan2 2x= x - -1 1
( ) p x = If rational function and p(x) polynomial with degree less than degree of the polynomial q(x), we factorized the denominator q(x) to linear factors or quadratic factors. ( ) f x ( ) q x
( ) p x A + B + = + + + ( )( ) ( ) ( ) x a x b x a x b A + ( ) p x B + C + = + + + 3 2 3 ( ) ( ) ( ) ( ) x a x a x a x a + A x + B + C ( ) p + x = + + + + 2 2 ( ( ) c ) ( ) ( ) ax bx g x h ax bx c g x h + + 2 where must not be factorized further, and A, B, C are constants must be evaluated. ax bx c
