Definite Integration · Mathematics · WB JEE

$$\int\limits_{ - \pi /2}^{\pi /2} {{{\sin }^9}x{{\cos }^5}x\,dx} $$ equals

If $$I = \int\limits_{ - \pi }^\pi {{{{e^{\sin x}}} \over {{e^{\sin x}} + {e^{ - \sin x}}}}dx} $$, then I equals

If $$h(x) = \int\limits_0^x {{{\sin }^4}t\,dt} $$, then $$h(x + \pi )$$ equals

The value of the integral $$\int\limits_0^2 {|{x^2} - 1|dx} $$ is

The value of $$\int\limits_0^\pi {|\cos x|dx} $$ is

The value of $$\int\limits_{ - 3}^3 {(a{x^5} + b{x^3} + cx + k)dx} $$, where a, b, c, k are constants, depends only on

The value of the integral $$\int\limits_{ - a}^a {{{x{e^{{x^2}}}} \over {1 + {x^2}}}dx} $$ is

The value of the $$\mathop {\lim }\limits_{n \to \infty } \left( {{1 \over {n + 1}} + {1 \over {n + 2}} + ... + {1 \over {6n}}} \right)$$ is

If $$f(x) = f(a - x)$$, then $$\int\limits_0^a {xf(x)dx} $$ is equal to

The value of $$\int\limits_0^\infty {{{dx} \over {({x^2} + 4)({x^2} + 9)}}} $$ is

If $${I_1} = \int\limits_0^{\pi /4} {{{\sin }^2}xdx} $$ and $${I_2} = \int\limits_0^{\pi /4} {{{\cos }^2}xdx} $$, then

$$\int\limits_{ - 1}^4 {f(x)dx = 4} $$ and $$\int\limits_2^4 {\{ 3 - f(x)\} dx = 7} $$, then the value of $$\int\limits_{ - 1}^2 {f(x)dx} $$ is

$$\int\limits_0^{1000} {{e^{x - [x]}}dx} $$ is equal to

The value of the integral $$\int\limits_0^{\pi /2} {{{\sin }^5}xdx} $$ is

If $${d \over {dx}}\{ f(x)\} = g(x)$$, then $$\int\limits_a^b {f(x)g(x)dx} $$ is equal to

If $${I_1} = \int\limits_0^{3\pi } {f({{\cos }^2}x)dx} $$ and $${I_2} = \int\limits_0^\pi {f({{\cos }^2}x)dx} $$, then

The value of $$I = \int\limits_{ - \pi /2}^{\pi /2} {|\sin x|dx} $$ is

If $$I = \int\limits_0^1 {{{dx} \over {1 + {x^{\pi /2}}}}} $$, then

The value of $$\int\limits_{ - 2}^2 {(x\cos x + \sin x + 1)dx} $$ is

$$\int\limits_\pi ^{16\pi } {|\sin x|dx = } $$

The value of $$\mathop {\lim }\limits_{n \to \infty } \sum\limits_{r = 1}^n {{{{r^3}} \over {{r^4} + {n^4}}}} $$ is

The value of $$\int\limits_0^\pi {{{\sin }^{50}}x{{\cos }^{49}}x\,dx} $$ is

All values of a for which the inequality $$\frac{1}{\sqrt{a}} \int_\limits1^a\left(\frac{3}{2} \sqrt{x}+1-\frac{1}{\sqrt{x}}\right) \mathrm{d} x...

For any integer $$\mathrm{n}, \int_\limits0^\pi \mathrm{e}^{\cos ^2 x} \cdot \cos ^3(2 n+1) x \mathrm{~d} x$$ has the value :

If $$\mathrm{f}(x)=\frac{\mathrm{e}^x}{1+\mathrm{e}^x}, \mathrm{I}_1=\int_\limits{\mathrm{f}(-\mathrm{a})}^{\mathrm{f}(\mathrm{a})} x \mathrm{~g}(x(1-...

Let $$f: \mathbb{R} \rightarrow \mathbb{R}$$ be a differentiable function and $$f(1)=4$$. Then the value of $$\lim _\limits{x \rightarrow 1} \int_\lim...

Let $$\mathrm{I}(\mathrm{R})=\int_\limits0^{\mathrm{R}} \mathrm{e}^{-\mathrm{R} \sin x} \mathrm{~d} x, \mathrm{R}>0$$. then,

$$\lim _\limits{n \rightarrow \infty} \frac{1}{n^{k+1}}[2^k+4^k+6^k+\ldots .+(2 n)^k]=$$

the expression $${{\int\limits_0^n {[x]dx} } \over {\int\limits_0^n {\{ x\} dx} }}$$, where $$[x]$$ and $$\{ x\} $$ are respectively integral and frac...

The value $$\int\limits_0^{1/2} {{{dx} \over {\sqrt {1 - {x^{2n}}} }}} $$ is $$(n \in N)$$

If $${I_n} = \int\limits_0^{{\pi \over 2}} {{{\cos }^n}x\cos nxdx} $$, then I$$_1$$, I$$_2$$, I$$_3$$ ... are in

$$\int\limits_0^{2\pi } {\theta {{\sin }^6}\theta \cos \theta d\theta } $$ is equal to

The average ordinate of $$y = \sin x$$ over $$[0,\pi ]$$ is :

Let f be derivable in [0, 1], then

The value of $$\int\limits_0^{{\pi \over 2}} {{{{{(\cos x)}^{\sin x}}} \over {{{(\cos x)}^{\sin x}} + {{(\sin x)}^{\cos x}}}}dx} $$ is

Let $$\mathop {\lim }\limits_{ \in \to 0 + } \int\limits_ \in ^x {{{bt\cos 4t - a\sin 4t} \over {{t^2}}}dt = {{a\sin 4x} \over x} - 1,\left( {0 ...

Let $$f(x) = \int\limits_{\sin x}^{\cos x} {{e^{ - {t^2}}}dt} $$. Then $$f'\left( {{\pi \over 4}} \right)$$ equals

If I is the greatest of $${I_1} = \int\limits_0^1 {{e^{ - x}}{{\cos }^2}x\,dx} $$, $${I_2} = \int\limits_0^1 {{e^{ - {x^2}}}{{\cos }^2}x\,dx} $$, $${I...

$$\int\limits_1^3 {{{\left| {x - 1} \right|} \over {\left| {x - 2} \right| + \left| {x - 3} \right|}}dx} $$ is equal to

The value of the integral $$\int\limits_{ - {1 \over 2}}^{{1 \over 2}} {{{\left\{ {{{\left( {{{x + 1} \over {x - 1}}} \right)}^2} + {{\left( {{{x - 1}...

If $$\int\limits_{{{\log }_e}2}^x {{{({e^x} - 1)}^{ - 1}}dx = {{\log }_e}{3 \over 2}} $$, then the value of x is

The value of $$\int\limits_0^5 {\max \{ {x^2},6x - 8\} \,dx} $$ is

Let f(x) be continuous periodic function with period T. Let $$I = \int\limits_a^{a + T} {f(x)\,dx} $$. Then

If $$b = \int\limits_0^1 {{{{e^t}} \over {t + 1}}dt} $$, then $$\int\limits_{a - 1}^a {{{{e^{ - t}}} \over {t - a - 1}}} $$ is

Let $$I = \int_{\pi /4}^{\pi /3} {{{\sin x} \over x}dx} $$. Then

The value of $$\sum\limits_{n = 1}^{10} {} \int\limits_{ - 2n - 1}^{ - 2n} {{{\sin }^{27}}} x\,dx + \sum\limits_{n = 1}^{10} {} \int\limits_{2n}^{2n +...

$$\int\limits_0^2 {[{x^2}]} \,dx$$ is equal to

Let f, be a continuous function in [0, 1], then $$\mathop {\lim }\limits_{n \to \infty } \sum\limits_{j = 0}^n {{1 \over n}} f\left( {{j \over n}} \ri...

The value of the integration $$\int\limits_{ - {\pi \over 4}}^{\pi /4} {\left( {\lambda |\sin x| + {{\mu \sin x} \over {1 + \cos x}} + \gamma } \righ...

The value of $$\mathop {\lim }\limits_{x \to 0} {1 \over x}\left[ {\int\limits_y^a {{e^{{{\sin }^2}t}}dt - } \int\limits_{x + y}^a {{e^{{{\sin }^2}t}}...

The value of the integral $$\int\limits_{ - 1}^1 {\left\{ {{{{x^{2015}}} \over {{e^{|x|}}({x^2} + \cos x)}} + {1 \over {{e^{|x|}}}}} \right\}} dx$$ is...

$$\mathop {\lim }\limits_{n \to \infty } {3 \over n}\left[ {1 + \sqrt {{n \over {n + 3}}} + \sqrt {{n \over {n + 6}}} + \sqrt {{n \over {n + 9}}} +...

If $$M = \int\limits_0^{\pi /2} {{{\cos x} \over {x + 2}}dx} $$, $$N = \int\limits_0^{\pi /4} {{{\sin x\cos x} \over {{{(x + 1)}^2}}}dx} $$, then the ...

The value of the integral $$I = \int_{1/2014}^{2014} {{{{{\tan }^{ - 1}}x} \over x}} dx$$ is

Let $$I = \int\limits_{\pi /4}^{\pi /3} {{{\sin x} \over x}} dx$$. Then

The value of $$I = \int_{\pi /2}^{5\pi /2} {{{{e^{{{\tan }^{ - 1}}(\sin x)}}} \over {{e^{{{\tan }^{ - 1}}(\sin x)}} + {e^{{{\tan }^{ - 1}}(\cos x)}}}}...

The value of $$\mathop {\lim }\limits_{n \to \infty } {1 \over n}\left\{ {{{\sec }^2}{\pi \over {4n}} + {{\sec }^2}{{2\pi } \over {4n}} + ... + {{\se...

Let $${I_1} = \int_0^n {[x]} \,dx$$ and $${I_2} = \int_0^n {\{ x\} } \,dx$$, where [x] and {x} are integral and fractional parts of x and n $$ \in $$ ...

The value of $$\mathop {\lim }\limits_{n \to \infty } \left[ {{n \over {{n^2} + {1^2}}} + {n \over {{n^2} + {2^2}}} + ... + {1 \over {2n}}} \right]$$ ...

The value of the integral $$\int_0^1 {{e^{{x^2}}}} dx$$

$$\int_0^{100} {{e^{x - [x]}}} dx$$ is equal to

If $$f(x) = \int_{ - 1}^x {|t|} \,dt$$, then for any $$x \ge 0,\,f(x)$$ is equal to

Let $$I = \int_0^{100\pi } {\sqrt {(1 - \cos 2x)} } \,dx$$, then

If m, n be integers, then find the value of $$\int\limits_{ - \pi }^\pi {{{(\cos mx - \sin nx)}^2}dx} $$

Evaluate the following integral $$\int\limits_{ - 1}^2 {|x\sin \pi x|dx} $$

Prove that $$I = \int\limits_0^{\pi /2} {{{\sqrt {\sec x} } \over {\sqrt {\cos ecx} + \sqrt {\sec x} }}dx = {\pi \over 4}} $$

$$ \text { The points of extremum of } \int_\limits0^{x^2} \frac{t^2-5 t+4}{2+e^t} d t \text { are } $$

Let f be a non-negative function defined on $$\left[ {0,{\pi \over 2}} \right]$$. If $$\int\limits_0^x {(f'(t) - \sin 2t)dt = \int\limits_x^0 {f(t)\t...

Which of the following statements are true?

Whichever of the following is/are correct?

Let $$f(x) = \left\{ {\matrix{ {0,} & {if} & { - 1 \le x \le 0} \cr {1,} & {if} & {x = 0} \cr {2,} & {if} & {0 ...

Let $${I_n} = \int\limits_0^1 {{x^n}} {\tan ^{ - 1}}xdx$$. If $${a_n}{I_{n + 2}} + {b_n}{I_n} = {c_n}$$ for all n $$ \ge $$ 1, then

Let $$I = \int\limits_0^I {{{{x^3}\cos 3x} \over {2 + {x^2}}}dx} $$, then

Let f be a non-constant continuous function for all x $$ \ge $$ 0. Let f satisfy the relation f(x) f(a $$-$$ x) = 1 for some a $$ \in $$ R+. Then, $$I...