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:
Step 1: Polynomial long division
The degree of the numerator is higher than that of the denominator , so we perform polynomial long division.
Division:
Now divide again:
Thus, the integral becomes:
Step 2: Decompose the remaining fraction
We perform partial fraction decomposition on:
Factor the denominator . Therefore:
Multiply through by :
Expand and group terms:
Equating coefficients:
Solve this system of equations:
From (1):
Substitute into (2):
Substitute into :
Thus:
Step 3: Rewrite the integral
The integral becomes:
Split into separate terms:
Step 4: Compute each term
Final Answer:
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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
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