Integral Comparison Analysis
We are asked to compare the values of three definite integrals:
- $I_1 = \int_0^1 e^{-x} \cos^2x \, dx$
- $I_2 = \int_0^1 e^{-x^2} \cos^2x \, dx$
- $I_3 = \int_0^1 e^{-x^2} \, dx$
The comparison depends on the integrands over the interval $[0, 1]$.
Comparing $I_3$ and $I_2$
Consider the integrands $f_3(x) = e^{-x^2}$ and $f_2(x) = e^{-x^2} \cos^2x$.
For any real value of $x$, we know that $0 \le \cos^2x \le 1$.
Multiplying by $e^{-x^2}$ (which is always positive), we get: $0 \le e^{-x^2} \cos^2x \le e^{-x^2}$
Integrating this inequality from $0$ to $1$: $\int_0^1 0 \, dx \le \int_0^1 e^{-x^2} \cos^2x \, dx \le \int_0^1 e^{-x^2} \, dx$
This simplifies to $0 \le I_2 \le I_3$.
Since $\cos^2x$ is not equal to 1 over the entire interval $[0, 1]$ (it's 0 at $x = \frac{\pi}{2} + n\pi$, and less than 1 elsewhere), the inequality is strict for the integral value, meaning $I_2 < I_3$.
Comparing $I_2$ and $I_1$
Consider the integrands $f_2(x) = e^{-x^2} \cos^2x$ and $f_1(x) = e^{-x} \cos^2x$.
We need to compare the terms $e^{-x^2}$ and $e^{-x}$ for $x \in [0, 1]$.
For $x$ in the interval $(0, 1)$, we have $x^2 < x$.
Multiplying by $-1$ reverses the inequality: $-x^2 > -x$.
Since the exponential function $y \mapsto e^y$ is strictly increasing, we have $e^{-x^2} > e^{-x}$ for $x \in (0, 1)$.
Given that $\cos^2x \ge 0$, we can multiply the inequality by $\cos^2x$: $e^{-x^2} \cos^2x > e^{-x} \cos^2x$ for $x \in (0, 1)$.
Integrating this inequality from $0$ to $1$: $\int_0^1 e^{-x^2} \cos^2x \, dx > \int_0^1 e^{-x} \cos^2x \, dx$
Therefore, $I_2 > I_1$.
Final Integral Ordering
Combining the results from the previous steps:
- $I_2 < I_3$
- $I_2 > I_1$
These inequalities together establish the order: $I_3 > I_2 > I_1$.
Similar Questions
- Let a differentiable function $f$ satisfy the equation $\int_{0}^{36} f\left(\frac{tx}{36}\right) dt = 4\alpha f(x)$. If $y = f(x)$ is a standard parabola passing through the points (2, 1) and (– 4, $\beta$), then $\beta^\alpha$ is equal to ______.
- Let $[\cdot]$ denote the greatest integer function and $f(x) = \lim_{n \to \infty} \frac{1}{n^3} \sum_{k=1}^n \left[ \frac{k^2}{3^x} \right]$. Then $12 \sum_{j=1}^\infty f(j)$ is equal to __________.
Let $f$ be a differentiable function satisfying $f(x) = 1 - 2x + \int_{0}^{x} e^{(x - t)} f(t) dt$, $x \in \mathbf{R}$ and let $g(x) = \int_{0}^{x} (f(t) + 2)^{15} (t - 4)^{6} (t + 12)^{17} dt$, $x \in \mathbf{R}$. If $\text{p}$ and $\text{q}$ are respectively the points of local minima and local maxima of $g$, then the value of $|\text{p} + \text{q}|$ is equal to _________.
Let $f$ be a twice differentiable function such that $f(x) = \int_{0}^{x} \tan(t - x)dt - \int_{0}^{x} f(t)\tan t dt$, $x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$.
Then $f''\left(\frac{\pi}{6}\right) + 12f'\left(-\frac{\pi}{6}\right) + f\left(\frac{\pi}{6}\right)$ is equal to __________- Let $y = y(x)$ be the solution of the differential equation $\left(x^2 - x\sqrt{x^2 - 1}\right)dy + \left(y\left(x - \sqrt{x^2 - 1}\right) - x\right)dx = 0, x \geq 1$. If $y(1) = 1$, then the greatest integer less than $y\left(\sqrt{5}\right)$ is _________.
- If $\int \frac{(\sqrt{1+x^2}+x)^{10}}{(\sqrt{1+x^2}-x)^9} dx = \frac{1}{m} \left( (\sqrt{1+x^2}+x)^n (n\sqrt{1+x^2}-x) \right) + C$ where $C$ is the constant of integration and $m, n \in N$, then $m+n$ is equal to
Let [.] denote the greatest integer function. If $\int_{0}^{e^3} \left[\frac{1}{e^{x-1}}\right] dx = \alpha - \log_e 2$, then $\alpha^3$ is equal to ____________.
Let $f: R\to R$ be a thrice differentiable odd function satisfying $f'(x)\ge0, f''(x)=f(x), f(0)=0, f'(0)=3$. Then $9f(\log_e 3)$ is equal to ___________.
- Let $y = y (x)$ be the solution curve of the differentialequation $x (x^2 + e^x) dy + (e^x (x-2) y-x^3) dx = 0, x > 0$, passing through the point $(1, 0)$.Then $y (2)$ is equal to
$4\int_{0}^{1} (\frac{1}{\sqrt{3+x^2} + \sqrt{1+x^2}}) dx - 3\log_e (\sqrt{3})$ is equal to :
Important Questions from Integral Calculus
- Let a differentiable function $f$ satisfy the equation $\int_{0}^{36} f\left(\frac{tx}{36}\right) dt = 4\alpha f(x)$. If $y = f(x)$ is a standard parabola passing through the points (2, 1) and (– 4, $\beta$), then $\beta^\alpha$ is equal to ______.
- Let $[\cdot]$ denote the greatest integer function and $f(x) = \lim_{n \to \infty} \frac{1}{n^3} \sum_{k=1}^n \left[ \frac{k^2}{3^x} \right]$. Then $12 \sum_{j=1}^\infty f(j)$ is equal to __________.
Let $f$ be a differentiable function satisfying $f(x) = 1 - 2x + \int_{0}^{x} e^{(x - t)} f(t) dt$, $x \in \mathbf{R}$ and let $g(x) = \int_{0}^{x} (f(t) + 2)^{15} (t - 4)^{6} (t + 12)^{17} dt$, $x \in \mathbf{R}$. If $\text{p}$ and $\text{q}$ are respectively the points of local minima and local maxima of $g$, then the value of $|\text{p} + \text{q}|$ is equal to _________.
Let $f$ be a twice differentiable function such that $f(x) = \int_{0}^{x} \tan(t - x)dt - \int_{0}^{x} f(t)\tan t dt$, $x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$.
Then $f''\left(\frac{\pi}{6}\right) + 12f'\left(-\frac{\pi}{6}\right) + f\left(\frac{\pi}{6}\right)$ is equal to __________- Let $y = y(x)$ be the solution of the differential equation $\left(x^2 - x\sqrt{x^2 - 1}\right)dy + \left(y\left(x - \sqrt{x^2 - 1}\right) - x\right)dx = 0, x \geq 1$. If $y(1) = 1$, then the greatest integer less than $y\left(\sqrt{5}\right)$ is _________.
