Evaluate the following integrals. ∫(12 y-8)/(y^4-2 y^2+1) d...

Key Concepts

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its simpler factors. In integrals, it is often beneficial to factor expressions in the denominator or numerator to simplify the integrand, especially when such factors reveal structures like repeated or perfect square factors, which can lead to easier simplifications.

Partial Fraction Decomposition

Partial fraction decomposition is a method used to split complex rational expressions into a sum of simpler fractions whose integrals are more easily computed. This technique is particularly useful when the denominator can be factored into linear or quadratic factors, allowing the integral to be expressed in terms of standard forms.

Integration of Rational Functions

The integration of rational functions often requires combining algebraic techniques such as factoring and partial fractions to simplify the expression before integration. Recognizing the structure of the numerator and denominator, and reducing the fraction to a sum of simpler terms, allows the use of standard integration methods for computing antiderivatives.

Substitution Method

The substitution method is a strategy for simplifying integrals by making a change of variable. It can transform a complicated integrand into a simpler one by reparameterizing variables, which is particularly effective when parts of the integrand are derivatives of other parts, enabling a more straightforward integration process.

Transcript

00:01 In problem 36 we have to evaluate the integral 12 y minus 8 divided by y power 4 minus 2 y square plus 1 b y for this consider the integral which is 12 y minus 8 divided by 12 y minus 8 divided by y power 4 minus 2 y square plus 1 this can be written as 12 y minus 8 divided by here the denominator the expression in the denominator can be written as y square minus 1 4 or it can be written as y square minus 1 4 or it can be written as 12 y minus 8 divided by since y square minus 1 is y minus 1 into y plus 1 so it can be written as y minus 1 whole square into y plus 1 whole scale so we have two repeated linear factors now we decompose it in partial fractions that is let 12 y minus 8 divided by y minus 1 square into y plus 1 square is equal to a divided by y minus 1 plus b divided by y minus 1 square plus b divided by y minus 1 square y plus 1 plus d divided by y plus 1 squared.

02:42 Let us call it equation 1.

02:45 Now multiplying throughout by y minus 1 hore square into y plus 1 hod we get 12 y minus 8 is equal to a into y minus 1 times plus 1 squared plus b into y plus 1 squared plus d into y plus 1 squared plus c into y plus 1 1 into y minus 1 squared plus d into y minus 1 squared let us call it equation 2 now if we put y is equal to 1 in equation 2 we get 4 is equal to if if y is equal to 1 then a times 1 minus 1 is 0 so the whole expression will be 0 plus 1 plus 1 2 2 2 squared 4 times b plus the next 2 expression will be 0 so this implies that we get b is equal to 1 now if we put y is equal to minus 1 in equation 2 we will get minus 20 is equal to the first 3 expressions will be 0 and from the last expression we get 4d this implies that d is equal to minus 5.

04:59 Now to find the remaining two constants, we use comparing coefficients method.

05:10 Now compare the coefficients of y -quot and the question terms on both sides of equation 2.

05:20 We get two equations on a comparing coefficients that is if we compare the coefficient of y cube then we get the equation 0 is equal to a plus c let's call it equation 3 now if we compare the coefficients of constant terms on both sides are equation 2 then we get minus 8 is equal to minus a plus b plus c plus d let's call it equation 4...

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