Explanation Given: 100 x d x ( x − 3 ) ( x 2 + 1 ) 2 Formula Used: Calculation: Using the partial fraction decomposition, 100 x d x ( x − 3 ) ( x 2 + 1 ) 2 = A x − 3 + B x + C ( x 2 + 1 ) + D x + E ( x 2 + 1 ) 2 On simplification, we get 100 x = A ( x 2 + 1 ) 2 + ( B x + C ) ( x − 3 ) ( x 2 + 1 ) + ( D x + E ) ( x − 3 ) Setting x = 3 will give us A = 3 Substituting the value of A = 3 , we get 100 x = ( B + 3 ) x 4 + ( C − 3 B ) x 3 + ( B − 3 C + D + 3 ) x 2 + ( C − 3 B − 3 D + E ) x + ( 3 − 3 C − 3 E ) Equating the coefficients of like powers, B + 3 = 0 . Thus B = − 3 C − 3 B = 0 . Thus C = − 9 3 − 3 C − 3 E = 0 . Thus E = 10 B − 3 C − 3 D + E = 100 .Thus D = − 30 Now substituting the above, we get 100 x ( x − 3 ) ( x 2 + 1 ) 2 = 3 x − 3 + ( − 3 x − 9 ) ( x 2 + 1 ) + ( − 30 x + 100 ) ( x 2 + 1 ) 2 ∫ 100 x d x ( x − 3 ) ( x 2 + 1 ) 2 = ∫ 3 x − 3 d x + ∫ ( − 3 x − 9 ) ( x 2 + 1 ) d x + ∫ ( − 30 x + 100 ) ( x 2 + 1 ) 2 d x = 3 ∫ d x x − 3 − 3 ∫ x d x ( x 2 + 1 ) − 9 ∫ d x ( x 2 + 1 ) − 30 ∫ x d x ( x 2 + 1 ) 2 + 10 ∫ d x ( x 2 + 1 ) 2 For the second and fourth integrals, we use substitution method...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

See similar textbooks

Concept explainers

Riemann Sum

Riemann Sums is a special type of approximation of the area under a curve by dividing it into multiple simple shapes like rectangles or trapezoids and is used in integrals when finite sums are involved. Figuring out the area of a curve is complex hence t…

Riemann Integral

Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.

Videos

Question

Chapter 8.5, Problem 42E

To determine

To evaluate the integral, $\frac{100 x d x}{(x - 3) {(x^{2} + 1)}^{2}}$ .

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 8.5, Problem 41E

Chapter 8.5, Problem 43E

Students have asked these similar questions

4х+2 Evaluate the integral dx. (Hint: Use Algebra to help further rewrite your (x² +1)(x-1) fractions).

Decide whether you can find each integral using the formulas and techniques you have studied so far. Explain. (a) ∫2dx / √(x2 + 4) (b) ∫dx/x√(x2 − 9)

Concept Check: Evaluate the following integrals: 1. f(x¹ - 5x³+6) dx dv 2./-(1-3)* do 3. fz¹ √/325-5dz

Chapter 8 Solutions

Calculus - Standalone book

Ch. 8.1 - Prob. 1PQ Ch. 8.1 - Prob. 2PQ Ch. 8.1 - Prob. 3PQ Ch. 8.1 - Prob. 1E Ch. 8.1 - Prob. 2E Ch. 8.1 - Prob. 3E Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - Prob. 6E Ch. 8.1 - Prob. 7E

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

To evaluate the integral, 100 x d x ( x − 3 ) ( x 2 + 1 ) 2 .

Solution Summary: The author explains how to evaluate the integral using the partial tion decomposition.

Concept explainers

Videos

To evaluate the integral, $\frac{100 x d x}{(x - 3) {(x^{2} + 1)}^{2}}$ .

Want to see the full answer?

Chapter 8 Solutions

To evaluate the integral, 100 x d x ( x − 3 ) ( x 2 + 1 ) 2 .

Solution Summary: The author explains how to evaluate the integral using the partial tion decomposition.

Concept explainers

Videos

To evaluate the integral, 100xdx(x−3)(x2+1)2.

Want to see the full answer?

Chapter 8 Solutions

To evaluate the integral, $\frac{100 x d x}{(x - 3) {(x^{2} + 1)}^{2}}$ .