If $f(x)$ is a quadratic in $x$ , then $\int\limits_0^1 {f(x) dx}$ is
If $f(x)$ is a quadratic in $x$ , then $\int\limits_0^1 {f(x) dx}$ is
- A
$\frac{1}{6}\left( {f\left( 0 \right) + 4f\left( {\frac{1}{2}} \right) + f(1)} \right)$
- B
$\frac{1}{6}\left( {4f\left( 0 \right) + f\left( {\frac{1}{2}} \right) + f(1)} \right)$
- C
$\frac{1}{6}\left( {f\left( 0 \right) + f\left( {\frac{1}{2}} \right) + 4f(1)} \right)$
- D
$\frac{1}{6}\left( {f\left( 0 \right) + f\left( {\frac{1}{2}} \right) + f(1)} \right)$
Similar Questions
Let $I_1 = \int\limits_0^{\frac{\pi }{2}} {{e^{ - {x^2}}}\sin (x)dx} $ ; $I_2 = \int\limits_0^{\frac{\pi }{2}} {{e^{ - {x^2}}}dx} $ ; $I_3 = \int\limits_0^{\frac{\pi }{2}} {{e^{ - {x^2}}}(1 + x)\,dx} $
and consider the statements
$I\,:$ $I_1 < I_2$
$II\,:$ $I_2 < I_3$
$III\,:$ $I_1 = I_3$
Which of the following is $(are)$ true?
Let $I_1 = \int\limits_0^{\frac{\pi }{2}} {{e^{ - {x^2}}}\sin (x)dx} $ ; $I_2 = \int\limits_0^{\frac{\pi }{2}} {{e^{ - {x^2}}}dx} $ ; $I_3 = \int\limits_0^{\frac{\pi }{2}} {{e^{ - {x^2}}}(1 + x)\,dx} $
and consider the statements
$I\,:$ $I_1 < I_2$
$II\,:$ $I_2 < I_3$
$III\,:$ $I_1 = I_3$
Which of the following is $(are)$ true?
If $b _{ n }=\int \limits_{0}^{\frac{\pi}{2}} \frac{\cos ^{2} nx }{\sin x } dx , n \in N$, then
If $b _{ n }=\int \limits_{0}^{\frac{\pi}{2}} \frac{\cos ^{2} nx }{\sin x } dx , n \in N$, then
- [JEE MAIN 2022]
If $f (x) = \int\limits_0^\pi {\frac{{t\,\sin t\,dt}}{{\sqrt {1 + {{\tan }^2}x\,\,\,{{\sin }^2}t} }}\,} $ for $0 < x < \frac{\pi }{2}\,$
Let the function $f :[0,2] \rightarrow R$ be defined as
$f(x)=\left\{\begin{array}{cc}e^{\min \left[x^2, x-[x]\right\}}, & x \in[0,1) \\e^{\left[x-\log _e x\right]}, & x \in[1,2]\end{array}\right.$
where [t] denotes the greatest integer less than or equal to $t$. Then the value of the integral $\int \limits_0^2 x f(x) d x$ is
Let the function $f :[0,2] \rightarrow R$ be defined as
$f(x)=\left\{\begin{array}{cc}e^{\min \left[x^2, x-[x]\right\}}, & x \in[0,1) \\e^{\left[x-\log _e x\right]}, & x \in[1,2]\end{array}\right.$
where [t] denotes the greatest integer less than or equal to $t$. Then the value of the integral $\int \limits_0^2 x f(x) d x$ is
- [JEE MAIN 2023]
Let $f$ be a positive function. Let
${I_1} = \int_{1 - k}^k {x\,f\left\{ {x(1 - x)} \right\}} \,dx$, ${I_2} = \int_{1 - k}^k {\,f\left\{ {x(1 - x)} \right\}} \,dx$
when $2k - 1 > 0.$ Then ${I_1}/{I_2}$ is
Let $f$ be a positive function. Let
${I_1} = \int_{1 - k}^k {x\,f\left\{ {x(1 - x)} \right\}} \,dx$, ${I_2} = \int_{1 - k}^k {\,f\left\{ {x(1 - x)} \right\}} \,dx$
when $2k - 1 > 0.$ Then ${I_1}/{I_2}$ is
- [IIT 1997]