Differential Equations
Differential Equations
Allen Star Batch
Grade 12
Question:
If $f(x) = \int_1^x \frac{\log t}{1 + t + t^2} dt, x \geq 1$, then:
$f(x) = \int_1^x \frac{t \log t}{1 + t + t^2} dt, x \geq 1$
$f(x) = \int_1^{1/x} \frac{\log t}{1 + t + t^2} dt, x \geq 1$
$f(x) = f\left(\frac{1}{x}\right)$
All of these
Step-by-Step Solution
Key Concept: Substitution of reciprocal variables can linearize complex rational differential equations.
Let $t = \frac{1}{y}$. The substitution transforms the equation into $\int_1^{1/x} \frac{-ny}{1 + \frac{1}{y} + \frac{1}{y^2}} \cdot (-\frac{1}{y^2})\,dy = \int_1^x \frac{tny}{y^2 + y + 1}\,dy$. This integral evaluates to express the solution in terms of $y$ and $x$.
Correct Answer: 2,3