enEnglishguગુજરાતી
JEE Mains · Maths · STD 12 - 7.2 definite integral
If \({I_1} = \int\limits_0^1 {{e^{ - x}}} {\cos ^2}\,x\,dx\,;\,{I_2} = \int\limits_0^1 {{e^{ - {x^2}}}} {\cos ^2}\,x\,dx\) and \(\,{I_3} = \int\limits_0^1 {{e^{ - {x^3}}}} dx\) ; then
- A \(I_2>I_3>I_1\)
- B \(I_3>I_1>I_2\)
- C \(I_2>I_1>I_3\)
- D \(I_3>I_2>I_1\)
Answer & Solution
Correct Answer
(D) \(I_3>I_2>I_1\)
Step-by-step Solution
Detailed explanation
Given: \(I_{1}=\int_{0}^{1} e^{-x} \cos ^{2} x d x\) \(I_{2}=\int_{0}^{1} e^{-x^{2}} \cos ^{2} x d x\) and \(I_{3}=\int_{0}^{1} e^{-x^{3}} d x\) For \(x \in(0,1)\) \(\Rightarrow x>x^{2}\) or \(-x<-x^{2}\) and \(x^{2}>x^{3}\) or \(-x^{2}<-x^{3}\)…
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- For a \(3 \times 3\) matrix \(M\), let trace \((M)\) denote the sum of all the diagonal elements of \(M\). Let \(A\) be a \(3 \times 3\) matrix such that \(|A|=\frac{1}{2}\) and trace \((A)=3\). If \(B=\operatorname{adj}(\operatorname{adj}(2 A))\), then the value of \(|B|+\) trace (B) equals :JEE Mains 2025 Medium
- Let \(f : N \rightarrow R\) be a function such that \(f(x+y)=2 f(x) f(y)\) for natural numbers \(x\) and \(y\). If \(f(1)=2\), then the value of \(\alpha\) for which \(\sum \limits_{k=1}^{10} f(\alpha+k)=\frac{512}{3}\left(2^{20}-1\right)\) holds, isJEE Mains 2022 Hard
- If the tangent to the curve \(y\, = \,\frac{x}{{{x^2}\, - \,3}},\,x\, \in \,R,\,(x\, \ne \, \pm \,\sqrt 3 )\) at a point \((\alpha ,\,\beta )\,\ne\,(0,0)\) on it is parallel to the line \(2x + 6y -11 = 0\) thenJEE Mains 2019 Hard
- The number of solutions of the equation \(|\cot x|=\cot x+\frac{1}{\sin x}\) in the interval \([0,2 \pi]\) isJEE Mains 2021 Hard
- The number of values of \(z \in \mathbb{C}\), satisfying the equations \(|z-(4+8i)|=\sqrt{10}\) and \(|z-(3+5i)|+|z-(5+11i)|=4\sqrt{5}\), is:JEE Mains 2026 Hard
- Let \(A=\{0,1,2,3,4,5\}\). Let \(R\) be a relation on A defined by \((x, y) \in R\) if and only if max \(\{x, y\} \in\{3,4\}\). Then among the statements \(\left(\mathrm{S}_1\right)\) : The number of elements in R is 18 , and \(\left(\mathrm{S}_2\right)\) : The relation R is symmetric but neither reflexive nor transitiveJEE Mains 2025 Medium
More PYQs from JEE Mains
- If \(\vec{a}\) is nonzero vector such that its projections on the vectors \(2 \hat{i}-\hat{j}+2 \hat{k}, \hat{i}+2 \hat{j}-2 \hat{k}\) and \(\hat{k}\) are equal, then a unit vector along \(\vec{a}\) is:JEE Mains 2025 Medium
- \(\mathop {\lim }\limits_{x \to 0} \,\frac{{{{\sin }^2}\,x}}{{\sqrt 2 - \sqrt {1 + \cos \,x} }}\) equalsJEE Mains 2019 Hard
- For \(10\) observations \(x_1, x_2, \ldots, x_{10}\), if \(\sum_{i=1}^{10}(x_i+2)^2=180\) and \(\sum_{i=1}^{10}(x_i-1)^2=90\), then their standard deviation is:JEE Mains 2026 Medium
- A survey shows that \(73 \%\) of the persons working in an office like coffee, whereas \(65 \%\) like tea. If \(x\) denotes the percentage of them, who like both coffee and tea, then \(x\) cannot beJEE Mains 2020 Medium
- If \(\alpha, \beta \in R\) are such that \(1-2 i\) (here \(i ^{2}=-1\) ) is a root of \(z^{2}+\alpha z+\beta=0,\) then \((\alpha-\beta)\) is equal to ..... .JEE Mains 2021 Medium
- If the coefficients of \(x^4, x^5\) and \(x^6\) in the expansion of \((1+x)^n\) are in the arithmetic progression, then the maximum value of \(n\) is :JEE Mains 2024 Hard