Definite Integration
579 Questions
1995
JEE Advanced
MCQ
IIT-JEE 1995 Screening
The value of $\int\limits_\pi ^{2\pi } {\left[ {2\,\sin x} \right]\,dx} $ where [ . ] represents the greatest integer function is
A.
${{ - 5\pi } \over 3}$
B.
$\pi $
C.
${{ 5\pi } \over 3}$
D.
$ - 2\pi $
1994
JEE Advanced
Numerical
IIT-JEE 1994
Show that $\int\limits_0^{n\pi + v} {\left| {\sin x} \right|dx = 2n + 1 - \cos \,v} $ where $n$ is a positive integer and $\,0 \le v < \pi .$
Correct Answer: $$2n + 1 - \cos \gamma $$
1994
JEE Advanced
Numerical
IIT-JEE 1994
The value of $\int\limits_2^3 {{{\sqrt x } \over {\sqrt {3 - x} + \sqrt x }}} dx$ is ...........
Correct Answer: $${\raise0.5ex\hbox{$\scriptstyle 1$}
\kern-0.1em/\kern-0.15em
\lower0.25ex\hbox{$\scriptstyle 2$}}$$
1993
JEE Advanced
Numerical
IIT-JEE 1993
Evaluate $\int_2^3 {{{2{x^5} + {x^4} - 2{x^3} + 2{x^2} + 1} \over {\left( {{x^2} + 1} \right)\left( {{x^4} - 1} \right)}}} dx.$
Correct Answer: $${1 \over 6}\log 6 - {1 \over {10}}$$
1993
JEE Advanced
MCQ
IIT-JEE 1993
The value of $\int\limits_0^{\pi /2} {{{dx} \over {1 + {{\tan }^3}\,x}}} $ is
A.
$0$
B.
$1$
C.
$\pi /2$
D.
$\pi /4$
1993
JEE Advanced
Numerical
IIT-JEE 1993
The value of $\int\limits_{\pi /4}^{3\pi /4} {{\phi \over {1 + \sin \phi }}d\phi } $ is ..............
Correct Answer: $$\pi \left( {\sqrt 2 - 1} \right)$$
1992
JEE Advanced
Numerical
IIT-JEE 1992
Determine a positive integer $n \le 5,$ such that
$$\int\limits_0^1 {{e^x}{{\left( {x - 1} \right)}^n}dx = 16 - 6e} $$
Correct Answer: $$n = 3$$
1991
JEE Advanced
Numerical
IIT-JEE 1991
Evaluate $\,\int\limits_0^\pi {{{x\,\sin \,2x\,\sin \left( {{\pi \over 2}\cos x} \right)} \over {2x - \pi }}dx} $
Correct Answer: $${8 \over {{\pi ^2}}}$$
1990
JEE Advanced
Numerical
IIT-JEE 1990
Prove that for any positive integer $k$,
${{\sin 2kx} \over {\sin x}} = 2\left[ {\cos x + \cos 3x + ......... + \cos \left( {2k - 1} \right)x} \right]$
Hence prove that $\int\limits_0^{\pi /2} {\sin 2kx\,\cot \,x\,dx = {\pi \over 2}} $
${{\sin 2kx} \over {\sin x}} = 2\left[ {\cos x + \cos 3x + ......... + \cos \left( {2k - 1} \right)x} \right]$
Hence prove that $\int\limits_0^{\pi /2} {\sin 2kx\,\cot \,x\,dx = {\pi \over 2}} $
Correct Answer: Solve it.
1990
JEE Advanced
Numerical
IIT-JEE 1990
Show that $\int\limits_0^{\pi /2} {f\left( {\sin 2x} \right)\sin x\,dx = \sqrt 2 } \int\limits_0^{\pi /4} {f\left( {\cos 2x} \right)\cos x\,dx} $
Correct Answer: Solve it.
1990
JEE Advanced
MCQ
IIT-JEE 1990
Let $f:R \to R$ and $\,\,g:R \to R$ be continuous functions. Then the value of the integral
$\int\limits_{ - \pi /2}^{\pi /2} {\left[ {f\left( x \right) + f\left( { - x} \right)} \right]\left[ {g\left( x \right) - g\left( { - x} \right)} \right]dx} $ is
$\int\limits_{ - \pi /2}^{\pi /2} {\left[ {f\left( x \right) + f\left( { - x} \right)} \right]\left[ {g\left( x \right) - g\left( { - x} \right)} \right]dx} $ is
A.
$\pi $
B.
$1$
C.
$-1$
D.
$0$
1989
JEE Advanced
Numerical
IIT-JEE 1989
If $f$ and $g$ are continuous function on $\left[ {0,a} \right]$ satisfying
$f\left( x \right) = f\left( {a - x} \right)$ and $g\left( x \right) + g\left( {a - x} \right) = 2,$
then show that $\int\limits_0^a {f\left( x \right)g\left( x \right)dx = \int\limits_0^a {f\left( x \right)dx} } $
$f\left( x \right) = f\left( {a - x} \right)$ and $g\left( x \right) + g\left( {a - x} \right) = 2,$
then show that $\int\limits_0^a {f\left( x \right)g\left( x \right)dx = \int\limits_0^a {f\left( x \right)dx} } $
Correct Answer: Solve it.
1989
JEE Advanced
Numerical
IIT-JEE 1989
The value of $\int\limits_{ - 2}^2 {\left| {1 - {x^2}} \right|dx} $ is ...............
Correct Answer: $$4$$
1988
JEE Advanced
Numerical
IIT-JEE 1988
Evaluate $\int\limits_0^1 {\log \left[ {\sqrt {1 - x} + \sqrt {1 + x} } \right]dx} $
Correct Answer: $${1 \over 2}\left[ {\log 2 + {\pi \over 2} - 1} \right]$$
1988
JEE Advanced
Numerical
IIT-JEE 1988
The integral $\int\limits_0^{1.5} {\left[ {{x^2}} \right]dx,} $
Where [ ] denotes the greatest integer function, equals .............
Correct Answer: $$2 - \sqrt 2 $$
1988
JEE Advanced
MCQ
IIT-JEE 1988
The value of the integral $\int\limits_0^{2a} {[{{f\left( x \right)} \over {\left\{ {f\left( x \right) + f\left( {2a - x} \right)} \right\}}}]\,dx} $ is equal to $a$.
A.
TRUE
B.
FALSE
1987
JEE Advanced
Numerical
IIT-JEE 1987
$f\left( x \right) = \left| {\matrix{
{\sec x} & {\cos x} & {{{\sec }^2}x + \cot x\cos ec\,x} \cr
{{{\cos }^2}x} & {{{\cos }^2}x} & {\cos e{c^2}x} \cr
1 & {{{\cos }^2}x} & {{{\cos }^2}x} \cr
} } \right|.$
Then $\int\limits_0^{\pi /2} {f\left( x \right)dx = .......} $
Then $\int\limits_0^{\pi /2} {f\left( x \right)dx = .......} $
Correct Answer: $$ - \left( {{{15\pi + {{32}^ \circ }} \over {60}}} \right)$$
1986
JEE Advanced
Numerical
IIT-JEE 1986
Evaluate : $\int\limits_0^\pi {{{x\,dx} \over {1 + \cos \,\alpha \,\sin x}},0 < \alpha < \pi } $
Correct Answer: $${{\pi \alpha } \over {\sin x}}$$
1985
JEE Advanced
Numerical
IIT-JEE 1985
Evaluate the following : $\,\,\int\limits_0^{\pi /2} {{{x\sin x\cos x} \over {{{\cos }^4}x + {{\sin }^4}x}}} dx$
Correct Answer: $${{{\pi ^2}} \over {16}}$$
1985
JEE Advanced
MCQ
IIT-JEE 1985
For any integer $n$ the integral ...........
$\int\limits_0^\pi {{e^{{{\cos }^2}x}}{{\cos }^3}\left( {2n + 1} \right)xdx} $ has the value
$\int\limits_0^\pi {{e^{{{\cos }^2}x}}{{\cos }^3}\left( {2n + 1} \right)xdx} $ has the value
A.
$\pi $
B.
$1$
C.
$0$
D.
none of these
1984
JEE Advanced
Numerical
IIT-JEE 1984
Given a function $f(x)$ such that
(i) it is integrable over every interval on the real line and
(ii) $f(t+x)=f(x),$ for every $x$ and a real $t$, then show that
the integral $\int\limits_a^{a + 1} {f\,\,\left( x \right)} \,dx$ is independent of a.
(i) it is integrable over every interval on the real line and
(ii) $f(t+x)=f(x),$ for every $x$ and a real $t$, then show that
the integral $\int\limits_a^{a + 1} {f\,\,\left( x \right)} \,dx$ is independent of a.
Correct Answer: Solve it.
1984
JEE Advanced
Numerical
IIT-JEE 1984
Evaluate the following $\int\limits_0^{{1 \over 2}} {{{x{{\sin }^{ - 1}}x} \over {\sqrt {1 - {x^2}} }}dx} $
Correct Answer: $${{6 - \pi \sqrt 3 } \over {12}}$$
1983
JEE Advanced
Numerical
IIT-JEE 1983
Evaluate : $\int\limits_0^{\pi /4} {{{\sin x + \cos x} \over {9 + 16\sin 2x}}dx} $
Correct Answer: $${1 \over {20}}\log 3$$
1983
JEE Advanced
MCQ
IIT-JEE 1983
The value of the integral $\int\limits_0^{\pi /2} {{{\sqrt {\cot x} } \over {\sqrt {\cot x} + \sqrt {\tan x} }}dx} $ is
A.
$\pi /4$
B.
$\pi /2$
C.
$\pi $
D.
none of these
1982
JEE Advanced
Numerical
IIT-JEE 1982
Show that $\int\limits_0^\pi {xf\left( {\sin x} \right)dx} = {\pi \over 2}\int\limits_0^\pi {f\left( {\sin x} \right)dx.} $
Correct Answer: Solve it.
1982
JEE Advanced
Numerical
IIT-JEE 1982
Find the value of $\int\limits_{ - 1}^{3/2} {\left| {x\sin \,\pi \,x} \right|\,dx} $
Correct Answer: $${3 \over \pi } + {1 \over {{\pi ^2}}}$$
1981
JEE Advanced
Numerical
IIT-JEE 1981
Show that : $\mathop {\lim }\limits_{n \to \infty } \left( {{1 \over {n + 1}} + {1 \over {n + 2}} + .... + {1 \over {6n}}} \right) = \log 6$
Correct Answer: Solve it.
1981
JEE Advanced
MCQ
IIT-JEE 1981
Let $a, b, c$ be non-zero real numbers such that
$\int\limits_0^1 {\left( {1 + {{\cos }^8}x} \right)\left( {a{x^2} + bx + c} \right)dx = \int\limits_0^2 {\left( {1 + {{\cos }^8}x} \right)\left( {a{x^2} + bx + c} \right)dx.} } $
Then the quadratic equation $a{x^2} + bx + c = 0$ has
$\int\limits_0^1 {\left( {1 + {{\cos }^8}x} \right)\left( {a{x^2} + bx + c} \right)dx = \int\limits_0^2 {\left( {1 + {{\cos }^8}x} \right)\left( {a{x^2} + bx + c} \right)dx.} } $
Then the quadratic equation $a{x^2} + bx + c = 0$ has
A.
no root in $(0, 2)$
B.
at least one root in $(0, 2)$
C.
a double root in $(0, 2)$
D.
two imaginary roots
1981
JEE Advanced
MCQ
IIT-JEE 1981
The value of the definite integral $\int\limits_0^1 {\left( {1 + {e^{ - {x^2}}}} \right)} \,\,dx$
A.
$-1$
B.
$2$
C.
$1 + {e^{ - 1}}$
D.
none of these