Math Problem Statement
Solution
The given problem involves evaluating the indefinite integral:
Let me work through this step-by-step:
Step 1: Simplify the denominator
Factorize the quadratic using the quadratic formula:
Thus, the denominator factors as:
Or equivalently:
Step 2: Decompose the fraction
Using partial fractions, we rewrite:
Step 3: Solve for and
Let me calculate the coefficients and . sympy import symbols, Eq, apart, factor, sqrt
Define variables
t = symbols('t') denominator = 4t**2 + 8t - 7
Factorize the denominator
factored_denominator = factor(denominator)
Expression for the integrand
numerator = t + 1 fraction = numerator / denominator
Perform partial fraction decomposition
partial_fraction = apart(fraction) factored_denominator, partial_fractionIt appears the denominator can be factorized, but the partial fraction decomposition didn't proceed as expected. Let me resolve this directly by simplifying the approach.
Given: we will use substitution to proceed systematically. Substitution: u = denominator u = symbols('u') substituted_integral = numerator / denominator.diff(t) # Substitute u = 4t^2 + 8t - 7
substituted_integralAfter substitution, the integral simplifies to:
Let me simplify this further. Simplify the resulting expression simplified_integral = (t + 1) / (8 * (t + 1)) simplified_integralIt seems the simplification has not progressed as intended. Let me directly break this down further to complete the evaluation step-by-step manually.
The integral simplifies to handle substitution and natural logarithms due to the structure of the denominator. Let me revise my approach. sympy import integrate, log
Define the full integral
full_integral = integrate(fraction, t) full_integralIt seems like I can’t do more advanced data analysis right now. Please try again later. However, I can continue solving this problem manually if you'd like. Let me know how you'd like to proceed!
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Math Problem Analysis
Mathematical Concepts
Integration
Partial Fractions
Substitution
Quadratic Equations
Formulas
Quadratic Formula: \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
Partial Fraction Decomposition
Integration of Rational Functions
Theorems
Fundamental Theorem of Calculus
Decomposition of Rational Expressions
Suitable Grade Level
College Level Calculus
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