JEE Mains · Maths · STD 12 - 7.2 definite integral
Let f be a twice differentiable non-negative function such that \( (f(x))^{2}=25+\int_{0}^{x}((f(t))^{2}+(f'(t))^{2})dt \). Then the mean of \(f\left(\log _e(1)\right), f\left(\log _e(2)\right), \ldots \ldots, f\left(\log _e(625)\right)\) is equal to:
- A 1560
- B 1565
- C 1570
- D 1575
Answer & Solution
Correct Answer
(B) 1565
Step-by-step Solution
Detailed explanation
\(2 f(x) f^{\prime}(x)=f^2(x)+\left(f^{\prime}(x)\right)^2\) \(\Rightarrow\left(f(x)-f^{\prime}(x)\right)^2=0\) \(\Rightarrow f ( x )= f ^{\prime}( x )\) \(\Rightarrow \ell n ( f ( x ))= x + c \Rightarrow f ( x )= c ^{\prime} e ^{ x }\) \(f(0)=5 \Rightarrow f(x)=5 e^x\) Mean…
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
- Let \(Q\) be the mirror image of the point \(P (1,0,1)\) with respect to the plane \(S : x + y + z =5\). If a line \(L\) passing through \((1,-1,-1)\), parallel to the line \(PQ\) meets the plane \(S\) at \(R\), then \(QR ^{2}\) is equal toJEE Mains 2022 Hard
- The sum of the infinite series \(1+\frac{5}{6}+\frac{12}{6^{2}}+\frac{22}{6^{3}}+\frac{35}{6^{4}}+\frac{51}{6^{5}}+\frac{70}{6^{6}}+\ldots .\) is equal toJEE Mains 2022 Hard
- Let a curve \(y=f(x), x \in(0, \infty)\) pass through the points \(P\left(1, \frac{3}{2}\right)\) and \(Q\left(a, \frac{1}{2}\right)\). If the tangent at any point \(R(b, f(b))\) to the given curve cuts the \(y\)-axis at the point \(S(0, c)\) such that \(b c=3\), then \((P Q)^2\) is equal to \(.........\).JEE Mains 2023 Hard
- If the sum of all the roots of the equation \(e^{2 x}-11 e^{x}-45 e^{-x}+\frac{81}{2}=0\) is \(\log _{ e } P\), then \(p\) is equal toJEE Mains 2022 Hard
- Let \(O\) be the origin and the position vector of the point \(P\) be \(-\hat{i}-2 \hat{j}+3 \hat{k}\). If the position vectors of the points \(A , B\) and \(C\) are \(-2 \hat{i}+\hat{j}-3 \hat{k}, 2 \hat{i}+4 \hat{j}-2 \hat{k}\) and \(-4 \hat{i}+2 \hat{j}-\hat{k}\) respectively then the projection of the vector \(\overline{O P}\) on a vector perpendicular to the vectors \(\overline{A B}\) and \(\overline{A C}\) is \(......\).JEE Mains 2023 Hard
- If \(5\left( {{{\tan }^2}x - {{\cos }^2}x} \right) = 2\cos 2x + 9,\) then \(\cos 4x\) is equal toJEE Mains 2017 Hard
More PYQs from JEE Mains
- If an unbiased die, marked with \(-2,-1,0,1,2,3\) on its faces, is through five times, then the probability that the product of the outcomes is positive, is :JEE Mains 2023 Hard
- If \(\int_{0}^{\sqrt{3}} \frac{15 x^{3}}{\sqrt{1+x^{2}+\sqrt{\left(1+x^{2}\right)^{3}}}} d x=\alpha \sqrt{2}+\beta \sqrt{3}\), where \(\alpha, \beta\) are integers, then \(\alpha+\beta\) is equal to.JEE Mains 2022 Hard
- Let \(\vec{a}=6 \hat{i}+\hat{j}-\hat{k}\) and \(\vec{b}=\hat{i}+\hat{j}\). If \(\vec{c}\) is a is vector such that \(|\vec{c}| \geq 6, \vec{a} \cdot \vec{c}=6|\vec{c}|,|\vec{c}-\vec{a}|=2 \sqrt{2}\) and the angle between \(\vec{a} \times \vec{b}\) and \(\vec{c}\) is \(60^{\circ}\), then \(|(\vec{a} \times \vec{b}) \times \vec{c}|\) is equal to :JEE Mains 2024 Hard
- Let \(f ( x )=\left[2 x ^{2}+1\right]\) and \(g ( x )=\left\{\begin{array}{ll}2 x -3, & x < 0 \\ 2 x +3, & x \geq 0\end{array}\right.\), where \([t]\) is the greatest integer \(\leq t\) છે. Then, in the open interval \((-1,1)\), the number of points where fog is discontinuous is equal toJEE Mains 2022 Medium
- If for \(n \geq 1\) , \({P_n} = \int\limits_1^e {{{\left( {\log \,x} \right)}^n}\,dx} \) , then \(P_{10} - 90P_8\) is equal toJEE Mains 2014 Hard
- Let \(\quad S=\left\{z \in C-\{i, 2 i\}: \frac{z^2+8 i z-15}{z^2-3 i z-2} \in R \right\}\). \(\alpha-\frac{13}{11} i \in S , \alpha \in R -\{0\}\), then \(242 \alpha^2\) is equal toJEE Mains 2023 Hard