Calculate. ∫(d x)/(x^4+16)

Name: Problem 26, Chapter 8: Calculus: One and Several Variables
Uploaded: 2021-10-19T17:17:19-08:00
Duration: 13 min 16 s
Channel: Harmender Singh Yadav
Description: Calculate. ∫(d x)/(x^4+16)

Key Concepts

Polynomial Factorization

Polynomial factorization involves expressing a polynomial as a product of its irreducible factors. In integration, factorizing the denominator can be a critical step to enable partial fraction decomposition, especially when dealing with high-degree polynomials that do not immediately suggest a substitution.

Partial Fraction Decomposition

Partial fraction decomposition is a method used to break down a complex rational function into a sum of simpler fractions. This technique is especially useful when the denominator can be factored into lower degree polynomials, making it easier to integrate each term separately.

Integration

Integration is the process of finding the antiderivative or the area under a curve defined by a function. In the context of solving integrals, there are various techniques available to handle complex functions, and choosing the right method is crucial for obtaining a solution.

Rational Functions

A rational function is the ratio of two polynomials. Integrals involving rational functions often require specialized techniques because the algebraic structure of the denominator and numerator allows for methods like partial fractions or substitution to simplify the integration process.

Transcript

00:01 In this question, we need to integrate the given expression, which is 1 over x to the power 4 plus 16.

00:08 We can write this fraction 1 over x to the power 4 plus 16 as after factizing this, we will get 1 over x squared plus root 8x plus 4 times x squared minus root 8 8.

00:32 X plus 4 so we can write this fraction as the sum of partial fractions now this one over x to the power 4 plus 16 can be written as a x plus b over x square plus root 8x plus 4 plus cx plus d over x square minus root 8x plus 4 now adding these two fractions on the right hand side we will get 1 over x to the power 4 plus 16 equals to a x plus b times x plus b times x squared x square minus root 8 x plus 4 plus cx plus d times x square plus root 8x plus 4.

01:55 Over and by multiplying this we will get here as x to the power 4 plus 16 now we can see the denominator in the both of these cases is same on the left -hand side and right -hand side so we can equate the numerator and here this will be a x plus b times x square minus root 8x plus 4 plus cx plus d times x square plus root a d x plus 4 now comparing the coefficients and by multiplying these factors and comparing the coefficients of different terms having the different powers of x we will get the values of a b c and d we got the value of a as 1 over 16 root 2 the value of b as 1 over 8 value of c over 1 over 16 root 2 sorry minus 1 over 16 root 2 and the value of d as 1 by 8 so the value of d is 1 by 8 now we can write the given fraction which is 1 over x to the power 4 plus 16 as the sum of partial fraction that is a x plus b over x square plus root 8 x plus 4 so we will get it as a is 1 over 16 root 2 so this will be 1 over 16 root 2 x plus 1 over 8 over this will be x square plus root 8 plus 4 and plus c x over d c is minus 1 by 16 root 2 so this will be minus 1 by 16 root 2 x plus 1 by 8 over x is square minus root 8 x plus 4.

04:35 Now we can integrate this expression as the integral of 1 over x to the power 4 plus 16 .dx will be equal to the here we can take out here as 1 over 16 root 2 common from both of these terms the both of these fractions so 1 over 16 root 2 common and the here as the integral of this will become x plus 2 root 2 on the denumerator x plus 2 root 2 over x squared root 8 x plus 4 dot dx and similarly from the second well fraction we can take out as 1 over 16 root 2 common so this will become 1 over 16 root 2 integral of here this will become minus x plus 2 root 2 over x square minus root 8x plus 4 dot d x now we can we can split these integrals and we can write this integral as i and we will split these integrals into two two parts of these two integrals so this will become one over 16 root two times here this will become integral of integral of integral of by taking as 1 over half of integral of 2x plus 2 root 2 and again this will become we have multiplied 2 to the numerator and divided by 2 so this will become 2x plus 4 root 2 but we are writing here 2x plus 2 root 2 and we will write again that will be 2 root 2 in 2 in second integral so this will become 2x plus 2 root 2 2 over x square plus root 8x plus 4 and plus here dot d x plus integral of 2 root 2 over x square plus root 8 x plus 4 dot d x so this one is the integral of this first part first integral only we have split it into two parts now the integral of this second part is we can take out minus 1 by 16 root 2 common and we will get inside this as minus 1 by 16 root 2 times 1 by 2 and here integral of this will become the integral of by multiplying and dividing the numerator this will be 2x and minus 2 root 2 over this will be x square plus square root 8x plus 4 and we will split this into two parts as this will be plus plus 2 root 2 over same x square plus root 8 x plus 4 so but here it is minus outside this bracket so we will it will be again in minus so so this will become minus 2 root 2 over x squared plus root 8x plus 4.

09:00 Here it will be.

09:01 Dx and again integrating this and dot dx...

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