Solve these problems:

Notes: ( Below I have attached Sample Answer , Like these done these works)

Qst:

MTH 203 Calculus II Integration Methods Integration Flowchart

The integration method should apply to one of the following types of integrands:

Rules basic rules antidifferentiation apply

It Contains composite function,(())

It is a product of functions, f(x) g(x)

It is a rational function, f(x) = P(x) Q(x)

It contains trigonometric functions

Test integrals: ( One of these )

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+ ,

+ ,

Sample Answers:

RATIONAL FUNCTION start W f(x)= Px G x x/0 Multiply both sides by the LCD, Distribute, Simplify and then solve. Check your answer by substituting answer for (x) to see if you obtain a true statement. After multiplying both sides by the LCD, we will be left with "A LINEAR EQUATION" to solve. Sometimes, we will be left with " A QUADRATIC EQUATION". >ENDStart V P x f( x ) G x Factor all denominators and determine the LCD. Identify the restrictions. Multiply both sides of the equation by the LCD. Distribute carefully and then simplify. Solve the resulting equation. {NOTE; If the result is a quadratic equation. Rewrite it in standard form, factor, and then set each factor equal to 0} Check for extraneous solutions. Always substitute into the original equation, or the factored equivalent. ENDId is a rational function, & co - Pes G G( ) Definition A rational function is a function that is a fraction where both the numerator and denominator are polynomials . That's, a rate of too polynomials P and G where the denominator is not O. where G #D Excample ; Solve 5- Solution : We fist make a note that x 7 0 and then muchrigg both sides by the LCD, 32(! 304. (3 - 2) = 3xx (*)+ multiply both Sides the LCD. Distribute . ] 15 - 26 = 3 foc = / 12 Simplify, and then Solve SC = 12.Cheds check your answer by substituting i2 for of to see it true statement .4 12 12 12 = 12 Answer: The solution is 12. After multiplying both sides of the previous example by the LCD, we were left with a linear equation to solve. This is not always the case; sometimes we will be left with a quadratic equation.Test Integral 320 + 11 do . X - x - 6 Using Partial fractions NOTE, Partial fractions are the fractions used for the decomposition of a rational Besession. Explanation The denominator can be factored as (( - 3 ) 6( + 2 ) , s A 326 + 11 I- 3 + B - *+2 = (x- 3 ) (x* + 2 ) A (( + 2 ) + B(X - 3) = 3xx+ 11 Ax + 2A + BX - 3B = 326 + 1/ ( A + B) xx + ( 2A - 3B ) = 3xx + 1\fThe partial fraction decomposition is thanks therefore , 4 DC - 3 x42 . This can be integrated using the rule = x-3 = 4 120 - 3) - In / act z / + C. The answer = 4 / / x-31 - In c+2) + C