LFM Pure and Mechanics

jee-main 2007 Q109 Integral Equation with Symmetry or Substitution View

Let $F ( x ) = f ( x ) + f \left( \frac { 1 } { x } \right)$, where $f ( x ) = \int _ { 1 } ^ { x } \frac { \log t } { 1 + t } d t$. Then $F ( e )$ equals
(1) $\frac { 1 } { 2 }$
(2) 0
(3) 1
(4) 2

jee-main 2012 Q73 Piecewise/Periodic Function Integration View

If the integral $\displaystyle\int_{0}^{10} \frac{\lfloor x \rfloor e^{x}}{e^{\lfloor x \rfloor}} dx = \alpha(e-1)$, then $\alpha$ is equal to (where $\lfloor x \rfloor$ denotes the greatest integer function)
(1) $\frac{1}{e-1}$
(2) $\frac{10}{e-1}$
(3) $\frac{e}{e-1}$
(4) $\frac{e^{10}-1}{e-1}$

jee-main 2013 Q84 Integral Equation with Symmetry or Substitution View

Statement-I: The value of the integral $\int_{\pi/6}^{\pi/3} \frac{dx}{1 + \sqrt{\tan x}}$ is equal to $\frac{\pi}{6}$. Statement-II: $\int_a^b f(x)\, dx = \int_a^b f(a + b - x)\, dx$.
(1) Statement-I is true; Statement-II is false.
(2) Statement-I is false; Statement-II is true.
(3) Statement-I is true; Statement-II is true; Statement-II is a correct explanation for Statement-I.
(4) Statement-I is true; Statement-II is true; Statement-II is not a correct explanation for Statement-I.

jee-main 2014 Q84 Definite Integral Evaluation (Computational) View

The integral $\int _ { 0 } ^ { \pi } \sqrt { 1 + 4 \sin ^ { 2 } \frac { x } { 2 } - 4 \sin \frac { x } { 2 } } \, d x$ equals
(1) $4 \sqrt { 3 } - 4$
(2) $4 \sqrt { 3 } - 4 - \frac { \pi } { 3 }$
(3) $\pi - 4$
(4) $\frac { 2 \pi } { 3 } - 4 - 4 \sqrt { 3 }$

jee-main 2015 Q70 Definite Integral Evaluation (Computational) View

The integral $\int_{\pi/4}^{3\pi/4} \frac{dx}{1 + \cos x}$ is equal to:
(1) $-1$
(2) $-2$
(3) $2$
(4) $4$

jee-main 2015 Q84 Integral Equation with Symmetry or Substitution View

The integral $\int _ { 2 } ^ { 4 } \frac { \log x ^ { 2 } } { \log x ^ { 2 } + \log ( 6 - x ) ^ { 2 } } d x$ is equal to
(1) 6
(2) 2
(3) 4
(4) 1

jee-main 2016 Q74 Definite Integral Evaluation (Computational) View

The integral $\int_{\pi/4}^{3\pi/4} \frac{dx}{1+\cos x}$ is equal to: (1) $-1$ (2) $-2$ (3) $2$ (4) $4$

jee-main 2016 Q85 Integral Equation with Symmetry or Substitution View

The value of the integral $\int _ { 4 } ^ { 10 } \frac { \left[ x ^ { 2 } \right] } { \left[ x ^ { 2 } - 28 x + 196 \right] + \left[ x ^ { 2 } \right] } d x$, where $[ x ]$ denotes the greatest integer less than or equal to $x$, is
(1) $\frac { 1 } { 3 }$
(2) 6
(3) 7
(4) 3

jee-main 2017 Q68 Accumulation Function Analysis View

If $f : \mathbb { R } \to \mathbb { R }$ is a differentiable function and $f ( 2 ) = 6$, then $\lim _ { x \to 2 } \int _ { 6 } ^ { f ( x ) } \frac { 2 t \, d t } { ( x - 2 ) }$ is:
(1) $2 f ^ { \prime } ( 2 )$
(2) $12 f ^ { \prime } ( 2 )$
(3) $0$
(4) $24 f ^ { \prime } ( 2 )$

jee-main 2017 Q83 Definite Integral Evaluation (Computational) View

The integral $\displaystyle\int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} \frac{dx}{1 + \cos x}$ is equal to
(1) $-2$
(2) $2$
(3) $4$
(4) $-1$

jee-main 2017 Q83 Definite Integral Evaluation (Computational) View

The integral $\int _ { \frac { \pi } { 12 } } ^ { \frac { \pi } { 4 } } \frac { 8 \cos 2 x } { ( \tan x + \cot x ) ^ { 3 } } d x$ equals
(1) $\frac { 13 } { 256 }$
(2) $\frac { 15 } { 64 }$
(3) $\frac { 13 } { 32 }$
(4) $\frac { 15 } { 128 }$

jee-main 2017 Q85 Definite Integral Evaluation (Computational) View

The integral $\int _ { \frac { \pi } { 4 } } ^ { \frac { 3 \pi } { 4 } } \frac { d x } { 1 + \cos x }$ is equal to:
(1) $- 1$
(2) $- 2$
(3) 2
(4) 4

jee-main 2018 Q84 Integral Equation with Symmetry or Substitution View

The values of $\int _ { - \frac { \pi } { 2 } } ^ { \frac { \pi } { 2 } } \frac { \sin ^ { 2 } x } { 1 + 2 ^ { x } } d x$ is
(1) $\frac { \pi } { 4 }$
(2) $\frac { \pi } { 8 }$
(3) $\frac { \pi } { 2 }$
(4) $4 \pi$

jee-main 2018 Q84 Integral Equation with Symmetry or Substitution View

The value of the integral $\int _ { - \frac { \pi } { 2 } } ^ { \frac { \pi } { 2 } } \sin ^ { 4 } x \left( 1 + \ln \left( \frac { 2 + \sin x } { 2 - \sin x } \right) \right) d x$ is
(1) $\frac { 3 } { 4 }$
(2) $\frac { 3 } { 8 } \pi$
(3) 0
(4) $\frac { 3 } { 16 } \pi$

jee-main 2018 Q84 Integral Equation with Symmetry or Substitution View

The value of the integral
$$\int _ { - \frac { \pi } { 2 } } ^ { \frac { \pi } { 2 } } \sin ^ { 4 } x \left( 1 + \log \left( \frac { 2 + \sin x } { 2 - \sin x } \right) \right) d x$$
is
(1) $\frac { 3 } { 16 } \pi$
(2) 0
(3) $\frac { 3 } { 8 } \pi$
(4) $\frac { 3 } { 4 }$

jee-main 2019 Q82 Definite Integral Evaluation (Computational) View

Let $f$ be a differentiable function from $R$ to $R$ such that $|f(x) - f(y)| \leq 2|x-y|^{3/2}$, for all $x,y \in R$. If $f(0) = 1$ then $\int_0^1 f^2(x)\,dx$ is equal to
(1) 0
(2) 1
(3) 2
(4) $\frac{1}{2}$

jee-main 2019 Q83 Definite Integral Evaluation (Computational) View

The integral $\int _ { 1 } ^ { e } \left\{ \left( \frac { x } { e } \right) ^ { 2 x } - \left( \frac { e } { x } \right) ^ { x } \right\} \log _ { e } x \, d x$ is equal to
(1) $\frac { 3 } { 2 } - e - \frac { 1 } { 2 e ^ { 2 } }$
(2) $\frac { 1 } { 2 } - e - \frac { 1 } { e ^ { 2 } }$
(3) $- \frac { 1 } { 2 } + \frac { 1 } { e } - \frac { 1 } { 2 e ^ { 2 } }$
(4) $\frac { 3 } { 2 } - \frac { 1 } { e } - \frac { 1 } { 2 e ^ { 2 } }$

jee-main 2019 Q83 Antiderivative Verification and Construction View

$\int \frac{\sin\frac{5x}{2}}{\sin\frac{x}{2}} dx$ is equal to
(1) $x + 2\sin x + \sin 2x + c$
(2) $2x + \sin x + \sin 2x + c$
(3) $x + 2\sin x + 2\sin 2x + c$
(4) $2x + \sin x + 2\sin 2x + c$

jee-main 2019 Q83 Definite Integral Evaluation (Computational) View

The value of the integral $\int _ { 0 } ^ { 1 } x \cot ^ { - 1 } \left( 1 - x ^ { 2 } + x ^ { 4 } \right) d x$ is
(1) $\frac { \pi } { 4 } - \frac { 1 } { 2 } \log _ { e } 2$
(2) $\frac { \pi } { 4 } - \log _ { e } 2$
(3) $\frac { \pi } { 2 } - \log _ { e } 2$
(4) $\frac { \pi } { 2 } - \frac { 1 } { 2 } \log _ { e } 2$

jee-main 2019 Q84 Integral Equation with Symmetry or Substitution View

If $f(x) = \frac{2 - x\cos x}{2 + x\cos x}$ and $g(x) = \log_e x$, then the value of the integral $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} g(f(x))\, dx$ is
(1) $\log_e e$
(2) $\log_e 2$
(3) $\log_e 1$
(4) $\log_e 3$

jee-main 2019 Q84 Integral Equation with Symmetry or Substitution View

The value of $\int _ { 0 } ^ { \pi / 2 } \frac { \sin ^ { 3 } x } { \sin x + \cos x } d x$ is:
(1) $\frac { \pi - 1 } { 2 }$
(2) $\frac { \pi - 2 } { 8 }$
(3) $\frac { \pi - 1 } { 4 }$
(4) $\frac { \pi - 2 } { 4 }$

jee-main 2019 Q84 Finding a Function from an Integral Equation View

If $f : R \rightarrow R$ is a differentiable function and $f ( 2 ) = 6$, then $\lim _ { x \rightarrow 2 } \int _ { 6 } ^ { f ( x ) } \frac { 2 t d t } { ( x - 2 ) }$ is:
(1) 0
(2) $2 f ^ { \prime } ( 2 )$
(3) $24 f ^ { \prime } ( 2 )$
(4) $12 f ^ { \prime } ( 2 )$

jee-main 2019 Q85 Finding a Function from an Integral Equation View

Let $f ( x ) = \int _ { 0 } ^ { x } g ( t ) \, dt$, where $g$ is a non-zero even function. If $f ( x + 5 ) = g ( x )$, then $\int _ { 0 } ^ { x } f ( t ) \, dt$ equals
(1) $\int _ { 5 } ^ { x + 5 } g ( t ) \, dt$
(2) $\int _ { x + 5 } ^ { 5 } g ( t ) \, dt$
(3) $5 \int _ { x + 5 } ^ { 5 } g ( t ) \, dt$
(4) $2 \int _ { 5 } ^ { x + 5 } g ( t ) \, dt$

jee-main 2020 Q65 Integral Equation with Symmetry or Substitution View

If $f(a + b + 1 - x) = f(x)$, for all $x$, where $a$ and $b$ are fixed positive real numbers, then $\frac { 1 } { a + b } \int _ { a } ^ { b } x (f(x) + f(x + 1)) d x$ is equal to
(1) $\int _ { a - 1 } ^ { b - 1 } f(x + 1) d x$
(2) $\int _ { a - 1 } ^ { b - 1 } f(x) d x$
(3) $\int _ { a + 1 } ^ { b + 1 } f(x) d x$
(4) $\int _ { a + 1 } ^ { b + 1 } f(x + 1) d x$

jee-main 2020 Q65 Definite Integral Evaluation (Computational) View

The value of $\alpha$ for which $4 \alpha \int _ { - 1 } ^ { 2 } e ^ { - \alpha | x | } d x = 5$, is
(1) $\log _ { e } 2$
(2) $\log _ { e } \left( \frac { 3 } { 2 } \right)$
(3) $\log _ { e } \sqrt { 2 }$
(4) $\log _ { e } \left( \frac { 4 } { 3 } \right)$

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LFM Pure and Mechanics

Indices and Surds 107

Exponential Functions 100

Laws of Logarithms 200

Exponential Equations & Modelling 58

Arithmetic Sequences and Series 271

Geometric Sequences and Series 203

Differentiation from First Principles 37

Tangents, normals and gradients 93

Stationary points and optimisation 384

Applied differentiation 164

Indefinite & Definite Integrals 533

Areas Between Curves 83

Areas by integration 109

Volumes of Revolution 61

Numerical integration 32

Vectors Introduction & 2D 204

Vectors 3D & Lines 233

Travel graphs 11

SUVAT in 2D & Gravity 6

Constant acceleration (SUVAT) 32

Variable acceleration (1D) 40

Variable acceleration (vectors) 26

Forces, equilibrium and resultants 9

Newton's laws and connected particles 18

Pulley systems 5

Motion on a slope 4

Friction 10

Momentum and Collisions 15

Moments 13

Parametric curves and Cartesian conversion 11

Parametric differentiation 15

Parametric integration 4

Projectiles 40