Math Problem Statement

Compute the following integral.

∫2x3−4x2−4x+3x2−2x−3dx=    + C.

FORMATTING: In Mobius, |a| is written "abs(a)".

FORMATTING: Do not include the constant of integration in your response.  It is already written for you.

Solution

To compute the given integral, we first rewrite the integrand and simplify the expression:

2x34x24x+3x22x3dx\int \frac{2x^3 - 4x^2 - 4x + 3}{x^2 - 2x - 3} \, dx

Step 1: Polynomial long division

The degree of the numerator 2x34x24x+32x^3 - 4x^2 - 4x + 3 is higher than that of the denominator x22x3x^2 - 2x - 3, so we perform polynomial long division.

Division:

2x34x24x+3x22x3=2x+2x28x+3x22x3\frac{2x^3 - 4x^2 - 4x + 3}{x^2 - 2x - 3} = 2x + \frac{-2x^2 - 8x + 3}{x^2 - 2x - 3}

Now divide again:

2x28x+3x22x3=2+14x+(3)x22x3\frac{-2x^2 - 8x + 3}{x^2 - 2x - 3} = -2 + \frac{-14x + (-3)}{x^2 - 2x - 3}

Thus, the integral becomes:

2x34x24x+3x22x3dx=(2x2+14x3x22x3)dx\int \frac{2x^3 - 4x^2 - 4x + 3}{x^2 - 2x - 3} \, dx = \int \left( 2x - 2 + \frac{-14x - 3}{x^2 - 2x - 3} \right) dx

Step 2: Decompose the remaining fraction

We perform partial fraction decomposition on:

14x3x22x3\frac{-14x - 3}{x^2 - 2x - 3}

Factor the denominator x22x3=(x3)(x+1)x^2 - 2x - 3 = (x - 3)(x + 1). Therefore:

14x3x22x3=Ax3+Bx+1\frac{-14x - 3}{x^2 - 2x - 3} = \frac{A}{x - 3} + \frac{B}{x + 1}

Multiply through by x22x3x^2 - 2x - 3:

14x3=A(x+1)+B(x3)-14x - 3 = A(x + 1) + B(x - 3)

Expand and group terms:

14x3=A(x)+A+B(x)3B-14x - 3 = A(x) + A + B(x) - 3B

14x3=(A+B)x+(A3B)-14x - 3 = (A + B)x + (A - 3B)

Equating coefficients:

  1. A+B=14A + B = -14
  2. A3B=3A - 3B = -3

Solve this system of equations:

From (1): A=14BA = -14 - B

Substitute into (2):

14B3B=3-14 - B - 3B = -3

144B=3-14 - 4B = -3

4B=11B=114-4B = 11 \quad \Rightarrow \quad B = -\frac{11}{4}

Substitute B=114B = -\frac{11}{4} into A=14BA = -14 - B:

A=14(114)=14+114=564+114=454A = -14 - \left(-\frac{11}{4}\right) = -14 + \frac{11}{4} = -\frac{56}{4} + \frac{11}{4} = -\frac{45}{4}

Thus:

14x3x22x3=454x3+114x+1\frac{-14x - 3}{x^2 - 2x - 3} = \frac{-\frac{45}{4}}{x - 3} + \frac{-\frac{11}{4}}{x + 1}

Step 3: Rewrite the integral

The integral becomes:

(2x2+454x3+114x+1)dx\int \left( 2x - 2 + \frac{-\frac{45}{4}}{x - 3} + \frac{-\frac{11}{4}}{x + 1} \right) dx

Split into separate terms:

2xdx2dx4541x3dx1141x+1dx\int 2x \, dx - \int 2 \, dx - \frac{45}{4} \int \frac{1}{x - 3} \, dx - \frac{11}{4} \int \frac{1}{x + 1} \, dx

Step 4: Compute each term

  1. 2xdx=x2\int 2x \, dx = x^2
  2. 2dx=2x\int 2 \, dx = 2x
  3. 1x3dx=lnx3\int \frac{1}{x - 3} \, dx = \ln|x - 3|
  4. 1x+1dx=lnx+1\int \frac{1}{x + 1} \, dx = \ln|x + 1|

Final Answer:

x22x454lnx3114lnx+1x^2 - 2x - \frac{45}{4} \ln|x - 3| - \frac{11}{4} \ln|x + 1|

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Calculus
Polynomial Division
Partial Fraction Decomposition
Integration of Rational Functions

Formulas

Polynomial Division Algorithm: Dividend = (Divisor × Quotient) + Remainder
Partial Fraction Decomposition
Integral of 1/(x - a): ∫1/(x - a)dx = ln|x - a|

Theorems

Fundamental Theorem of Algebra (used to factorize polynomials)
Logarithmic Integration Rule: ∫1/x dx = ln|x|

Suitable Grade Level

College-Level Calculus