Fundamental Theorem & Properties of Definite Integrals

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JEE Main 2015

LEVELJEE Main

The integral $\int_{2}^{4} \frac{l o g x ^{2}}{l o g x ^{2} + l o g ( 36 - 12 x + x ^{2} )} d x$ is equal to:

The integral $\int_{0}^{π} 1 + 4 sin^{2} \frac{x}{2} - 4 sin \frac{x}{2} d x$ equals:

(A)

43 - 4

(B)

43 - 4 - \frac{π}{3}

(C)

π - 4

(D)

\frac{2 π}{3} - 4 - 43

JEE Main 2013

LEVELBoard

Statement-1: The value of the integral $\int_{π /6}^{π /3} \frac{d x}{1 + t a n x}$ is equal to $π /6$ . Statement-2: $\int_{a}^{b} f (x) d x = \int_{a}^{b} f (a + b - x) d x$ .

(A)

Statement-1 is true; Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

(B)

Statement-1 is true; Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.

(C)

Statement-1 is true; Statement-2 is false.

(D)

Statement-1 is false; Statement-2 is true.

JEE Main 2012

LEVELJEE Main

If $g (x) = \int_{0}^{x} cos 4 t d t$ , then $g (x + π)$ equals

* Multiple Correct Options

(A)

\frac{g ( x )}{g ( π )}

(B)

g (x) + g (π)

(C)

g (x) - g (π)

(D)

g (x) \cdot g (π)

JEE Main 2010

LEVELJEE Main

Let $p (x)$ be a function defined on $R$ such that $p^{'} (x) = p^{'} (1 - x)$ , for all $x \in [0, 1], p (0) = 1$ and $p (1) = 41$ . Then $\int_{0}^{1} p (x) d x$ equals

41

JEE Main 2009

LEVELJEE Main

$\int_{0}^{π} [cot x] d x$ , where $[.]$ denotes the greatest integer function, is equal to:

- π /2

(D)

π /2

JEE Main 2007

LEVELBoard

The solution for $x$ of the equation $\int_{2}^{x} \frac{d t}{t t ^{2} - 1} = \frac{π}{2}$ is

(A)

\frac{3}{2}

(B)

22

$\int_{- π /2}^{π /2} [(x + π)^{3} + cos^{2} (x + 3 π)] d x$ is equal to

(A)

\frac{π ^{4}}{32}

(B)

\frac{π ^{4}}{32} + \frac{π}{2}

(C)

\frac{π}{2}

(D)

\frac{π}{4} - 1

JEE Main 2006

LEVELJEE Main

The value of $\int_{1}^{a} [x] f^{'} (x) d x, a > 1$ where $[x]$ denotes the greatest integer not exceeding $x$ is

(A)

a f (a) - {f (1) + f (2) + \dots + f ([a])}

(B)

[a] f (a) - {f (1) + f (2) + \dots + f ([a])}

(C)

[a] f ([a]) - {f (1) + f (2) + \dots + f (a)}

(D)

a f ([a]) - {f (1) + f (2) + \dots + f (a)}

JEE Main 2005

LEVELJEE Main

If $I_{1} = \int_{0}^{1} 2^{x^{2}} d x, I_{2} = \int_{0}^{1} 2^{x^{3}} d x, I_{3} = \int_{1}^{2} 2^{x^{2}} d x$ and $I_{4} = \int_{1}^{2} 2^{x^{3}} d x$ then

(A)

I_{2} > I_{1}

(B)

I_{1} > I_{2}

(C)

I_{3} = I_{4}

(D)

I_{3} > I_{4}

JEE Main 2005

LEVELJEE Main

The value of $\int_{- π}^{π} \frac{c o s ^{2} x}{1 + a ^{x}} d x, a > 0$ , is

(A)

aπ

(B)

π /2

(C)

π / a

(D)

2 π

JEE Main 2005

LEVELBoard

The value of integral, $\int_{3}^{6} \frac{x}{9 - x + x} d x$ is

The value of $\int_{- 2}^{3} ∣1 - x^{2} ∣ d x$ is

The value of $I = \int_{0}^{π /2} \frac{( s i n x + c o s x ) ^{2}}{1 + s i n 2 x} d x$ is

If $\int_{0}^{π} x f (sin x) d x = A \int_{0}^{π /2} f (sin x) d x$ , then $A$ is

(A)

2 π

(B)

π

(C)

π /4

(D)

JEE Main 2004

LEVELJEE Main

If $f (x) = \frac{e ^{x}}{1 + e ^{x}}$ , $I_{1} = \int_{f (- a)}^{f (a)} xg {x (1 - x)} d x$ and $I_{2} = \int_{f (- a)}^{f (a)} g {x (1 - x)} d x$ , then the value of $I_{2} / I_{1}$ is

Let $I = \int_{0}^{1} \frac{s i n x}{x} d x$ and $J = \int_{0}^{1} \frac{c o s x}{x} d x$ . Then which one of the following is true?

(A)

I > 2/3

and

J > 2

(B)

I < 2/3

and

J < 2

(C)

I < 2/3

and

J > 2

(D)

I > 2/3

and

J < 2

JEE Main 2003

LEVELJEE Main

If $f (y) = e^{y}, g (y) = y; y > 0$ and $F (t) = \int_{0}^{t} f (t - y) g (y) d y$ , then

(A)

F (t) = t e^{- t}

(B)

F (t) = 1 - t e^{- t} (1 + t)

(C)

F (t) = e^{t} - (1 + t)

(D)

F (t) = t e^{t}

JEE Main 2003

LEVELJEE Main

If $f (a + b - x) = f (x)$ then $\int_{a}^{b} x f (x) d x$ is equal to

(A)

\frac{a + b}{2} \int_{a}^{b} f (a + b + x) d x

(B)

\frac{a + b}{2} \int_{a}^{b} f (b - x) d x

(C)

\frac{a + b}{2} \int_{a}^{b} f (x) d x

(D)

\frac{b - a}{2} \int_{a}^{b} f (x) d x

JEE Main 2003

LEVELJEE Main

The value of the integral $I = \int_{0}^{1} x (1 - x)^{n} d x$ is

(A)

\frac{1}{n + 1} + \frac{1}{n + 2}

(B)

\frac{1}{n + 1}

(C)

\frac{1}{n + 2}

(D)

\frac{1}{n + 1} - \frac{1}{n + 2}

JEE Main 2002

LEVELJEE Main

$\int_{0}^{10 π} ∣ sin x ∣ d x$ is

$\int_{0}^{2} [x^{2}] d x$ is

(A)

2 - 2

(B)

2 + 2

(C)

2 - 1

(D)

5 - 2 - 3

JEE Main 2002

LEVELJEE Main

$\int_{- π}^{π} \frac{2 x ( 1 + s i n x )}{1 + c o s ^{2} x} d x$ is

(A)

π^{2} /4

(B)

π^{2}

(C)

zero

(D)

π /2

JEE Main 2002

LEVELJEE Main

If $y = f (x)$ makes +ve intercept of 2 and 0 unit on x and y axes and encloses an area of 3/4 square unit with the axes then $\int_{0}^{2} x f^{'} (x) d x$ is

The value of $\int_{- π /2}^{π /2} \frac{x ^{2} c o s x}{1 + e ^{x}} d x$ is equal to

(A)

\frac{π ^{2}}{4} - 2

(B)

\frac{π ^{2}}{4} + 2

(C)

π^{2} - e^{2}

(D)

π^{2} + e^{2}

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let I=∫01​x​sinx​dx and J=∫01​x​cosx​dx. Then which one of the following is true?

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The integral $\int_{2}^{4} \frac{l o g x ^{2}}{l o g x ^{2} + l o g ( 36 - 12 x + x ^{2} )} d x$ is equal to:

The integral $\int_{0}^{π} 1 + 4 sin^{2} \frac{x}{2} - 4 sin \frac{x}{2} d x$ equals:

Statement-1: The value of the integral $\int_{π /6}^{π /3} \frac{d x}{1 + t a n x}$ is equal to $π /6$ . Statement-2: $\int_{a}^{b} f (x) d x = \int_{a}^{b} f (a + b - x) d x$ .

If $g (x) = \int_{0}^{x} cos 4 t d t$ , then $g (x + π)$ equals

Let $p (x)$ be a function defined on $R$ such that $p^{'} (x) = p^{'} (1 - x)$ , for all $x \in [0, 1], p (0) = 1$ and $p (1) = 41$ . Then $\int_{0}^{1} p (x) d x$ equals

$\int_{0}^{π} [cot x] d x$ , where $[.]$ denotes the greatest integer function, is equal to:

The solution for $x$ of the equation $\int_{2}^{x} \frac{d t}{t t ^{2} - 1} = \frac{π}{2}$ is

$\int_{- π /2}^{π /2} [(x + π)^{3} + cos^{2} (x + 3 π)] d x$ is equal to

The value of $\int_{1}^{a} [x] f^{'} (x) d x, a > 1$ where $[x]$ denotes the greatest integer not exceeding $x$ is

If $I_{1} = \int_{0}^{1} 2^{x^{2}} d x, I_{2} = \int_{0}^{1} 2^{x^{3}} d x, I_{3} = \int_{1}^{2} 2^{x^{2}} d x$ and $I_{4} = \int_{1}^{2} 2^{x^{3}} d x$ then

The value of $\int_{- π}^{π} \frac{c o s ^{2} x}{1 + a ^{x}} d x, a > 0$ , is

The value of integral, $\int_{3}^{6} \frac{x}{9 - x + x} d x$ is

The value of $\int_{- 2}^{3} ∣1 - x^{2} ∣ d x$ is

The value of $I = \int_{0}^{π /2} \frac{( s i n x + c o s x ) ^{2}}{1 + s i n 2 x} d x$ is

If $\int_{0}^{π} x f (sin x) d x = A \int_{0}^{π /2} f (sin x) d x$ , then $A$ is

If $f (x) = \frac{e ^{x}}{1 + e ^{x}}$ , $I_{1} = \int_{f (- a)}^{f (a)} xg {x (1 - x)} d x$ and $I_{2} = \int_{f (- a)}^{f (a)} g {x (1 - x)} d x$ , then the value of $I_{2} / I_{1}$ is

Let $I = \int_{0}^{1} \frac{s i n x}{x} d x$ and $J = \int_{0}^{1} \frac{c o s x}{x} d x$ . Then which one of the following is true?

If $f (y) = e^{y}, g (y) = y; y > 0$ and $F (t) = \int_{0}^{t} f (t - y) g (y) d y$ , then

If $f (a + b - x) = f (x)$ then $\int_{a}^{b} x f (x) d x$ is equal to

The value of the integral $I = \int_{0}^{1} x (1 - x)^{n} d x$ is

$\int_{0}^{10 π} ∣ sin x ∣ d x$ is

$\int_{0}^{2} [x^{2}] d x$ is

$\int_{- π}^{π} \frac{2 x ( 1 + s i n x )}{1 + c o s ^{2} x} d x$ is

If $y = f (x)$ makes +ve intercept of 2 and 0 unit on x and y axes and encloses an area of 3/4 square unit with the axes then $\int_{0}^{2} x f^{'} (x) d x$ is

The value of $\int_{- π /2}^{π /2} \frac{x ^{2} c o s x}{1 + e ^{x}} d x$ is equal to