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Newton-Leibnitz's Formula - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

Quick Facts

  • Newton-Leibniz's Formula is considered one of the most asked concept.

  • 45 Questions around this concept.

Solve by difficulty

QuestionMEDIUM

\lim _{x \rightarrow 0} \frac{\int_0^{x^2} \cos t^2 d t}{x \sin x}=\ldots \ldots

 

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If F(x)=\frac{1}{x^2} \int_4^x\left\{4 t^2-2 F^{\prime}(t)\right\} d t, then F^{\prime}(4)  equals:

 

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\mathrm{ \lim _{x \rightarrow 0}\left(\frac{1}{x^5} \int_0^x c^{-t^2} d t-\frac{1}{x^4}+\frac{1}{3 x^2}\right)}

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If the function f:[0,8] \rightarrow R  is differentiable then for 0<\alpha, \beta<2, \int_0^8 f(t) d t

is equal to

View Solution

Concepts Covered - 1

Newton-Leibniz's Formula

If the functions u(x) and v(x) are defined and f(t) is a continuous function, then

\\\mathbf{\frac{d}{dx}\left [ \int_{u(x)}^{v(x)}f(t)\;dt \right ]=f(v(x))\cdot\frac{d}{dx}\left \{ v(x) \right \}-f(u(x))\cdot\frac{d}{dx}\left \{u(x) \right \}}

Proof:

\\\mathrm{Let\;\;\;\;\;\;\;\frac{d}{dx}\left \{ F(x) \right \}=f(x)}\\\\\mathrm{\Rightarrow \;\;\;\;\;\;\;\int_{u(x)}^{v(x)}f(t)\;dt=F(v(x))-F(u(x))}\\\\\\\mathrm{\Rightarrow \;\;\;\;\;\;\;\frac{d}{dx}\left [\int_{u(x)}^{v(x)}f(t)\;dt \right ]=\frac{d}{dx}\left (F(v(x))-F(u(x)) \right )}\\\\\\\mathrm{\Rightarrow \;\;\;\;\;\;\;\frac{d}{dx}\left [\int_{u(x)}^{v(x)}f(t)\;dt \right ]=F'(v(x))\frac{d}{dx}\left \{ v(x) \right \}-F'(u(x))\frac{d}{dx}\left \{ u(x) \right \}}\\\\\\\mathrm{\Rightarrow \;\;\;\;\;\;\;\frac{d}{dx}\left [\int_{u(x)}^{v(x)}f(t)\;dt \right ]=f(v(x))\frac{d}{dx}\left \{ v(x) \right \}-f(u(x))\frac{d}{dx}\left \{ u(x) \right \}}

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Newton-Leibniz's Formula

Study other Related Concepts

Newton-Leibnitz's Formula Current Topic

Integration as an Inverse Process of Differentiation
Integration of Trigonometric Functions
Integration by Substitution
Fundamental Formulae of Indefinite Integration (Inverse Trigonometric Functions)
Integrals of Particular Function
Trigonometric Integrals
Integration By Parts Formula
Integration Using Partial Fraction
Integration of Irrational Algebraic Function
Reduction Formula

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Reference Books

Newton-Leibniz's Formula

Mathematics for Joint Entrance Examination JEE (Advanced) : Calculus

Page No. : 8.27

Line : 27

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