JEE Mains · Maths · STD 12 - 7.2 definite integral
Let \([ x ]\) denote the greatest integer \(\leq x\). Consider the function \(f(x)=\max \left\{x^2, 1+[x]\right\}\). Then the value of the integral \(\int \limits_0^2 f ( x ) dx\) is :
- A \(\frac{5+4 \sqrt{2}}{3}\)
- B \(\frac{8+4 \sqrt{2}}{3}\)
- C \(\frac{1+5 \sqrt{2}}{3}\)
- D \(\frac{4+5 \sqrt{2}}{3}\)
Answer & Solution
Correct Answer
(A) \(\frac{5+4 \sqrt{2}}{3}\)
Step-by-step Solution
Detailed explanation
\(A=\int \limits_0^1 1 . d x+\int \limits_1^{\sqrt{2}} 2 d x+\int \limits_{\sqrt{2}}^2 x^2 d x\) \(=1+2 \sqrt{2}-2+\frac{8}{3}-\frac{2 \sqrt{2}}{3}\) \(=\frac{5}{3}+\frac{4 \sqrt{2}}{3}\)
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
- \(\begin{aligned}
& \text { If } \frac{1}{1^4}+\frac{1}{2^4}+\frac{1}{3^4}+\ldots . . \infty=\frac{\pi^4}{90}, \\
& \frac{1}{1^4}+\frac{1}{3^4}+\frac{1}{5^4}+\ldots . . \infty=\alpha, \\
& \frac{1}{2^4}+\frac{1}{4^4}+\frac{1}{6^4}+\ldots . \infty=\beta,
\end{aligned}\)
then \(\frac{\alpha}{\beta}\) is equal toJEE Mains 2025 Medium - Let \(A=\left[\begin{array}{cc}\alpha & -1 \\ 6 & \beta\end{array}\right], \alpha \gt 0\), such that \(\operatorname{det}(A)=0\) and \(\alpha+\beta=1\). If I denotes \(2 \times 2\) identity matrix, then the matrix \((1+\mathrm{A})^8\) is:JEE Mains 2025 Medium
- Let the mean of the data
be \(5.\) If \(m\) and \(\sigma^2\) are respectively the mean deviation about the mean and the variance of the data, then \(\frac{3 \alpha}{m+\sigma^2}\) is equal to \(..........\).\(X\) \(1\) \(3\) \(5\) \(7\) \(9\) \((f)\) \(4\) \(24\) \(28\) \(\alpha\) \(8\) JEE Mains 2023 Hard - The value of \(\int\limits_{0}^{2\pi } {\left[ {\sin \,2x\left( {1 + \cos \,3x} \right)} \right]} \,dx\) where \([t]\) denotes the greatest integer function, isJEE Mains 2019 Hard
- Let \(\alpha, \beta\) be the roots of the equation \(x^2 - 3x + r = 0\), and \(\dfrac{\alpha}{2}, 2\beta\) be the roots of the equation \(x^2 + 3x + r = 0\). If the roots of the equation \(x^2 + 6x = m\) are \(2\alpha + \beta + 2r\) and \(\alpha - 2\beta - \dfrac{r}{2}\), then \(m\) is equal to:JEE Mains 2026 Hard
- Let \(f(x)=\int_0^x\left(t+\sin \left(1-e^t\right)\right) d t, x \in \mathbb{R}\). Then \(\lim _{x \rightarrow 0} \frac{f(x)}{x^3}\) is equal toJEE Mains 2024 Hard
More PYQs from JEE Mains
- The number of integers, greater than \(7000\) that can be formed, using the digits \(3,5,6,7,8\) without repetition, isJEE Mains 2023 Medium
- A triangle is formed by the tangents at the point \((2,2)\) on the curves \(y^2=2 x\) and \(x^2+y^2=4 x\), and the line \(x+y+2=0\). If \(r\) is the radius of its circumcircle, then \(r ^2\) is equal to \(........\).JEE Mains 2023 Hard
- The shortest distance between the line \(\frac{x-3}{4}=\frac{y+7}{-11}=\frac{z-1}{5} \text { and } \frac{x-5}{3}=\frac{y-9}{-6}=\frac{z+2}{1}\) is :JEE Mains 2024 Medium
- Three distinct numbers are selected randomly from the set \(\{1,2,3, \ldots \ldots, 40\}\). If the probability, that the selected numbers are in an increasing G.P. is \(\frac{m}{n}\), \(\operatorname{gcd}(m, n)=1\), then \(m+n\) is equal to _____.JEE Mains 2025 Hard
- Let \(\vec{a}=6 \hat{i}+9 \hat{j}+12 \hat{k}, \vec{b}=\alpha \hat{i}+11 \hat{j}-2 \hat{k}\) and \(\vec{c}\) be vectors such that \(\vec{a} \times \vec{c}=\vec{a} \times \vec{b}\). If \(\vec{a} \cdot \vec{c}=-12\), \(\vec{c} .(\hat{i}-2 \hat{j}+\hat{k})=5\), then \(\vec{c} \cdot(\hat{i}+\hat{j}+\hat{k})\) is equal to \(.............\).JEE Mains 2023 Hard
- Consider a curve \(y=y(x)\) in the first quadrant as shown in the figure. Let the area \(A_{1}\) is twice the area \(A _{2}\). Then the normal to the curve perpendicular to the line \(2 x -12 y =15\) does NOT pass through the point.
JEE Mains 2022 Hard