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
- If $\int (\sin x)^{\frac{-11}{2}} (\cos x)^{\frac{-5}{2}} dx = - \frac{p_1}{q_1} (\cot x)^{\frac{9}{2}} - \frac{p_2}{q_2} (\cot x)^{\frac{5}{2}} - \frac{p_3}{q_3} (\cot x)^{\frac{1}{2}} + \frac{p_4}{q_4} (\cot x)^{\frac{-3}{2}} + C$, where $p_i$ and $q_i$ are positive integers with $\gcd(p_i, q_i) = 1$ for $i = 1, 2, 3, 4$ and C is the constant of integration, then $\frac{15 p_1 p_2 p_3 p_4}{q_1 q_2 q_3 q_4}$ is equal to _________
- 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 $f(\alpha)$ denote the area of the region in the first quadrant bounded by $x = 0, x = 1, y^2 = x$ and $y = |\alpha x - 5| - |1 - \alpha x| + \alpha x^2$. Then $(f(0) + f(1))$ is equal to
Let $f(x) = \int \frac{7x^{10} + 9x^8}{(1 + x^2 + 2x^9)^2} \,dx$, $x > 0$, $\lim_{x \rightarrow 0} f(x) = 0$ and $f(1) = \frac{1}{4}$.
If $A = \begin{bmatrix} 0 & 0 & 1 \\ \frac{1}{4} & f'(1) & 1 \\ \alpha^2 & 4 & 1 \end{bmatrix}$ and $B = \text{adj}(\text{adj } A)$ be such that $|B| = 81$, then $\alpha^2$ 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 $f : \mathbf{R} \rightarrow \mathbf{R}$ be a function such that $f(x) + 3f\left(\frac{\pi}{2} - x\right) = \sin x$, $x \in \mathbf{R}$. Let the maximum value of $f$ on $\mathbf{R}$ be $\alpha$. If the area of the region bounded by the curves $g(x) = x^{2}$ and $h(x) = \beta x^{3}, \beta > 0$, is $\alpha^{2}$, then $30\beta^{3}$ 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 $\int \frac{2x+5}{\sqrt{7-6x-x^2}} dx$ = $A\sqrt{7-6x-x^2} + Bsin^{-1} \left( \frac{x+3}{4} \right) + C$
(Where C is a constant of integration), then the ordered pair (A,B) is equal to :-
Important Questions from Integral Calculus
- If $\int (\sin x)^{\frac{-11}{2}} (\cos x)^{\frac{-5}{2}} dx = - \frac{p_1}{q_1} (\cot x)^{\frac{9}{2}} - \frac{p_2}{q_2} (\cot x)^{\frac{5}{2}} - \frac{p_3}{q_3} (\cot x)^{\frac{1}{2}} + \frac{p_4}{q_4} (\cot x)^{\frac{-3}{2}} + C$, where $p_i$ and $q_i$ are positive integers with $\gcd(p_i, q_i) = 1$ for $i = 1, 2, 3, 4$ and C is the constant of integration, then $\frac{15 p_1 p_2 p_3 p_4}{q_1 q_2 q_3 q_4}$ is equal to _________
- 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 $f(\alpha)$ denote the area of the region in the first quadrant bounded by $x = 0, x = 1, y^2 = x$ and $y = |\alpha x - 5| - |1 - \alpha x| + \alpha x^2$. Then $(f(0) + f(1))$ is equal to
Let $f(x) = \int \frac{7x^{10} + 9x^8}{(1 + x^2 + 2x^9)^2} \,dx$, $x > 0$, $\lim_{x \rightarrow 0} f(x) = 0$ and $f(1) = \frac{1}{4}$.
If $A = \begin{bmatrix} 0 & 0 & 1 \\ \frac{1}{4} & f'(1) & 1 \\ \alpha^2 & 4 & 1 \end{bmatrix}$ and $B = \text{adj}(\text{adj } A)$ be such that $|B| = 81$, then $\alpha^2$ 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 __________.
