Definite Integration
113 Questions
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