Math Problem Statement

Evaluate the indefinite integral \( \int \frac{t + 1}{4t^2 + 8t - 7} \, dt \).

Solution

The given problem involves evaluating the indefinite integral:

t+14t2+8t7dt\int \frac{t + 1}{4t^2 + 8t - 7} \, dt

Let me work through this step-by-step:

Step 1: Simplify the denominator 4t2+8t74t^2 + 8t - 7

Factorize the quadratic 4t2+8t74t^2 + 8t - 7 using the quadratic formula: t=b±b24ac2a,a=4,b=8,c=7t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \quad a = 4, b = 8, c = -7

t=8±824(4)(7)2(4)=8±64+1128=8±1768=8±4118=2±112t = \frac{-8 \pm \sqrt{8^2 - 4(4)(-7)}}{2(4)} = \frac{-8 \pm \sqrt{64 + 112}}{8} = \frac{-8 \pm \sqrt{176}}{8} = \frac{-8 \pm 4\sqrt{11}}{8} = \frac{-2 \pm \sqrt{11}}{2}

Thus, the denominator factors as: 4t2+8t7=4(t+2+112)(t+2112)4t^2 + 8t - 7 = 4 \left(t + \frac{2 + \sqrt{11}}{2}\right)\left(t + \frac{2 - \sqrt{11}}{2}\right)

Or equivalently: 4t2+8t7=4(t+1+112)(t+1112)4t^2 + 8t - 7 = 4 \left(t + 1 + \frac{\sqrt{11}}{2}\right)\left(t + 1 - \frac{\sqrt{11}}{2}\right)

Step 2: Decompose the fraction

Using partial fractions, we rewrite: t+14t2+8t7=At+1+112+Bt+1112\frac{t + 1}{4t^2 + 8t - 7} = \frac{A}{t + 1 + \frac{\sqrt{11}}{2}} + \frac{B}{t + 1 - \frac{\sqrt{11}}{2}}

Step 3: Solve for AA and BB

Let me calculate the coefficients AA and BB. 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 4t2+8t74t^2 + 8t - 7 can be factorized, but the partial fraction decomposition didn't proceed as expected. Let me resolve this directly by simplifying the approach.

Given: t+14t2+8t7dt\int \frac{t + 1}{4t^2 + 8t - 7} \, dt 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:

t+18t+8dt\int \frac{t + 1}{8t + 8} \, dt

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