Please show how to get to the answer. 4x+1+In\x + 2| + Cx*+2x34x282+ 1Z + 지미은 + 쭈-쭈+ 1지미=xptr

Name: Please show how to get to the answer. 4x+1+In\x + 2| + Cx*+2x34x282+ 1
Uploaded: 2024-03-20T07:07:37.000Z
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Description: Please show how to get to the answer. 4x+1+In\x + 2| + Cx*+2x34x282+ 1Z + 지미은 + 쭈-쭈+ 1지미=xptr

Given that: The integral is ∫4 x3 + 1x4 + 2x3= 18 ln x + 14x -14x2 +318 ln x+ 2 + C

Answered: 31 In |x| + + In]x + 2| + C 8 4x 4x2

Algebra: Structure And Method, Book 1

(REV)00th Edition

ISBN:9780395977224

Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Chapter1: Introduction To Algebra

Section1.6: Translating Problems Into Equations

Problem 6MRE

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Please show how to get to the answer.

4x+1
+In\x + 2| + C
x*+2x3
4x2
8
2+ 1Z + 지미은 + 쭈-쭈+ 1지미=xptr

Expert Solution

Step 1

Given that:

The integral is $\int \frac{4 x^{3} + 1}{x^{4} + 2 x^{3}} = \frac{1}{8} \ln |x| + \frac{1}{4 x} - \frac{1}{4 x^{2}} + \frac{31}{8} \ln |x + 2| + C$

Step 2

By using,

Partial decomposition formula,

$\int \frac{1}{x^{3} (x - a)} d x = \frac{A}{x} + \frac{B}{x^{2}} + \frac{C}{x^{3}} + \frac{D}{(x - a)} B y p a r t s r u l e, \int u d v = u v - \int v d u$

Step 3

$\int \frac{4 x^{3} + 1}{x^{4} + 2 x^{3}} d x = \int \frac{4 x^{3}}{x^{4} + 2 x^{3}} d x + \int \frac{1}{x^{4} + 2 x^{3}} d x \int \frac{4 x^{3}}{x^{4} + 2 x^{3}} d x = 4 \int \frac{x^{3}}{x^{4} + 2 x^{3}} d x A p p l y i n t e r g r a t i o n b y p a r t s, u = \frac{x^{3}}{x^{4} + 2 x^{3}}, d v = d x t h e n, v^{'} = 1 T h e n, 4 \int \frac{x^{3}}{x^{4} + 2 x^{3}} d x = 4 (\frac{x}{x + 2} . 1 - \int - \frac{x}{{(x + 2)}^{2}} d x) = 4 (\frac{x}{x + 2} + \ln |x + 2| + \frac{2}{x + 2}) T o g e t, 4 \int \frac{x^{3}}{x^{4} + 2 x^{3}} d x = 4 (\frac{x}{x + 2} + \ln |x + 2| + \frac{2}{x + 2})$

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