Definite Integration

The minimum value of the twice differentiable function \(f(x)=\int\limits_{0}^{x} \mathrm{e}^{x-\mathrm{t}} f^{\prime}(\mathrm{t}) \mathrm{dt}-\left(x^{2}-x+1\right) \mathrm{e}^{x}\), \(x \in \mathbf{R}\), is :

The value of the integral \(\int\limits_{0}^{\frac{\pi}{2}} 60 \frac{\sin (6 x)}{\sin x} d x\) is equal to _________.

Let \(I_{n}(x)=\int_{0}^{x} \frac{1}{\left(t^{2}+5\right)^{n}} d t, n=1,2,3, \ldots .\) Then :

The value of the integral \(\int\limits_{ - 2}^2 {{{|{x^3} + x|} \over {({e^{x|x|}} + 1)}}dx}\) is equal to :

The integral \(\int\limits_0^1 {{1 \over {{7^{\left[ {{1 \over x}} \right]}}}}dx}\), where [ . ] denotes the greatest integer function, is equal to

Let f be a differentiable function in \(\left( {0,{\pi \over 2}} \right)\). If \(\int\limits_{\cos x}^1 {{t^2}\,f(t)dt = {{\sin }^3}x + \cos x}\), then \({1 \over {\sqrt 3 }}f'\left( {{1 \over {\sqrt 3 }}} \right)\) is equal to

If m and n respectively are the number of local maximum and local minimum points of the function \(f(x) = \int\limits_0^{{x^2}} {{{{t^2} - 5t + 4} \over {2 + {e^t}}}dt}\), then the ordered pair (m, n) is equal to

Let a function \(f: \mathbb{R} \rightarrow \mathbb{R}\) be defined as :
$$f(x)= \begin{cases}\int\limits_{0}^{x}(5-|t-3|) d t, & x>4 \\ x^{2}+b x & , x \leq 4\end{cases}$$
where \(\mathrm{b} \in \mathbb{R}\). If \(f\) is continuous at $$x=4...

Let \(I=\int_{\pi / 4}^{\pi / 3}\left(\frac{8 \sin x-\sin 2 x}{x}\right) d x\). Then

Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a function defined as
\(f(x)=a \sin \left(\frac{\pi[x]}{2}\right)+[2-x], a \in \mathbb{R}\) where \([t]\) is the greatest integer less than or equal to \(t\). If $$\mathop {\lim }\limits_{x \t...

Let f be a differentiable function satisfying \(f(x)=\frac{2}{\sqrt{3}} \int\limits_{0}^{\sqrt{3}} f\left(\frac{\lambda^{2} x}{3}\right) \mathrm{d} \lambda, x>0\) and \(f(1)=\sqrt{3}\). If \(y=f(x)\) passes through the point \((\alpha, 6)\)...

Let \(f(x)=\min \{[x-1],[x-2], \ldots,[x-10]\}\) where [t] denotes the greatest integer \(\leq \mathrm{t}\). Then $$\int\limits_{0}^{10} f(x) \mathrm{d} x+\int\limits_{0}^{10}(f(x))^{2} \mathrm{~d} x+\int\limits_{0}^{10}|f(x)| \mathrm{d} x$...

\(\int\limits_{0}^{2}\left(\left|2 x^{2}-3 x\right|+\left[x-\frac{1}{2}\right]\right) \mathrm{d} x\), where [t] is the greatest integer function, is equal to :

Let \(f(x)=2+|x|-|x-1|+|x+1|, x \in \mathbf{R}\).
Consider
\((\mathrm{S} 1): f^{\prime}\left(-\frac{3}{2}\right)+f^{\prime}\left(-\frac{1}{2}\right)+f^{\prime}\left(\frac{1}{2}\right)+f^{\prime}\left(\frac{3}{2}\right)=2\)
$$(\mathrm{S} 2):...

The value of the integral \({{48} \over {{\pi ^4}}}\int\limits_0^\pi {\left( {{{3\pi {x^2}} \over 2} - {x^3}} \right){{\sin x} \over {1 + {{\cos }^2}x}}dx}\) is equal to __________.

Let f(x) = max {|x + 1|, |x + 2|, ....., |x + 5|}. Then \(\int\limits_{ - 6}^0 {f(x)dx}\) is equal to __________.

The integral \({{24} \over \pi }\int_0^{\sqrt 2 } {{{(2 - {x^2})dx} \over {(2 + {x^2})\sqrt {4 + {x^4}} }}}\) is equal to ____________.

If \(\mathrm{n}(2 \mathrm{n}+1) \int_{0}^{1}\left(1-x^{\mathrm{n}}\right)^{2 \mathrm{n}} \mathrm{d} x=1177 \int_{0}^{1}\left(1-x^{\mathrm{n}}\right)^{2 \mathrm{n}+1} \mathrm{~d} x\), then \(\mathrm{n} \in \mathbf{N}\) is equal to __________...

If \(a = \mathop {\lim }\limits_{n \to \infty } \sum\limits_{k = 1}^n {{{2n} \over {{n^2} + {k^2}}}}\) and \(f(x) = \sqrt {{{1 - \cos x} \over {1 + \cos x}}}\), \(x \in (0,1)\), then :

\(\int\limits_{0}^{20 \pi}(|\sin x|+|\cos x|)^{2} d x \text { is equal to }\)

The value of \(\int\limits_0^\pi {{{{e^{\cos x}}\sin x} \over {(1 + {{\cos }^2}x)({e^{\cos x}} + {e^{ - \cos x}})}}dx}\) is equal to:

The value of b > 3 for which \(12\int\limits_3^b {{1 \over {({x^2} - 1)({x^2} - 4)}}dx = {{\log }_e}\left( {{{49} \over {40}}} \right)}\), is equal to ___________.

If \({b_n} = \int_0^{{\pi \over 2}} {{{{{\cos }^2}nx} \over {\sin x}}dx,\,n \in N}\), then

$$
\begin{aligned}
&\text { If } \lim _{n \rightarrow \infty} \frac{(n+1)^{k-1}}{n^{k+1}}[(n k+1)+(n k+2)+\ldots+(n k+n)] \\
&=33 \cdot \lim _{n \rightarrow \infty} \frac{1}{n^{k+1}} \cdot\left[1^{k}+2^{k}+3^{k}+\ldots+n^{k}\right]
\end{ali...

For any real number \(x\), let \([x]\) denote the largest integer less than equal to \(x\). Let \(f\) be a real valued function defined on the interval \([-10,10]\) by $$f(x)=\left\{\begin{array}{l}x-[x], \text { if }[x] \text { is odd } \\...

Let \({a_n} = \int\limits_{ - 1}^n {\left( {1 + {x \over 2} + {{{x^2}} \over 3} + \,\,.....\,\, + \,\,{{{x^{n - 1}}} \over n}} \right)dx}\) for every n \(\in\) N. Then the sum of all the elements of the set {n \(\in\) N : an \(\in\) (2, 30...

Let \(f\) be a twice differentiable function on \(\mathbb{R}\). If \(f^{\prime}(0)=4\) and \(f(x) + \int\limits_0^x {(x - t)f'(t)dt = \left( {{e^{2x}} + {e^{ - 2x}}} \right)\cos 2x + {2 \over a}x}\), then \((2 a+1)^{5}\, a^{2}\) is equal t...

Let \([t]\) denote the greatest integer less than or equal to \(t\). Then the value of the integral \(\int_{-3}^{101}\left([\sin (\pi x)]+e^{[\cos (2 \pi x)]}\right) d x\) is equal to

$$\mathop {\lim }\limits_{n \to \infty } {1 \over {{2^n}}}\left( {{1 \over {\sqrt {1 - {1 \over {{2^n}}}} }} + {1 \over {\sqrt {1 - {2 \over {{2^n}}}} }} + {1 \over {\sqrt {1 - {3 \over {{2^n}}}} }} + \,\,...\,\, + \,\,{1 \over {\sqrt {1 - ...

Let \(\mathop {Max}\limits_{0\, \le x\, \le 2} \left\{ {{{9 - {x^2}} \over {5 - x}}} \right\} = \alpha\) and \(\mathop {Min}\limits_{0\, \le x\, \le 2} \left\{ {{{9 - {x^2}} \over {5 - x}}} \right\} = \beta\).
If $$\int\limits_{\beta - {...

Let \(f(\theta ) = \sin \theta + \int\limits_{ - \pi /2}^{\pi /2} {(\sin \theta + t\cos \theta )f(t)dt}\). Then the value of \(\left| {\int_0^{\pi /2} {f(\theta )d\theta } } \right|\) is _____________.

\(\mathop {\lim }\limits_{n \to \infty } \left( {{{{n^2}} \over {({n^2} + 1)(n + 1)}} + {{{n^2}} \over {({n^2} + 4)(n + 2)}} + {{{n^2}} \over {({n^2} + 9)(n + 3)}} + \,\,....\,\, + \,\,{{{n^2}} \over {({n^2} + {n^2})(n + n)}}} \right)\) is ...

The value of the integral \(\int\limits_{ - \pi /2}^{\pi /2} {{{dx} \over {(1 + {e^x})({{\sin }^6}x + {{\cos }^6}x)}}}\) is equal to

If \(x\phi (x) = \int\limits_5^x {(3{t^2} - 2\phi '(t))dt}\), x > \(-\)2, and \(\phi\)(0) = 4, then \(\phi\)(2) is __________.

Let [t] denote the greatest integer \(\le\) t. Then the value of \(8.\int\limits_{ - {1 \over 2}}^1 {([2x] + |x|)dx}\) is ___________.

Let f be a non-negative function in [0, 1] and twice differentiable in (0, 1). If \(\int_0^x {\sqrt {1 - {{(f'(t))}^2}} dt = \int_0^x {f(t)dt} }\), \(0 \le x \le 1\) and f(0) = 0, then $$\mathop {\lim }\limits_{x \to 0} {1 \over {{x^2}}}\i...

If [x] is the greatest integer \(\le\) x, then \({\pi ^2}\int\limits_0^2 {\left( {\sin {{\pi x} \over 2}} \right)(x - [x]} {)^{[x]}}dx\) is equal to :

Let \(F:[3,5] \to R\) be a twice differentiable function on (3, 5) such that \(F(x) = {e^{ - x}}\int\limits_3^x {(3{t^2} + 2t + 4F'(t))dt}\). If \(F'(4) = {{\alpha {e^\beta } - 224} \over {{{({e^\beta } - 4)}^2}}}\), then \(\alpha\) + $$\b...

Let the domain of the function\(f(x) = {\log _4}\left( {{{\log }_5}\left( {{{\log }_3}(18x - {x^2} - 77)} \right)} \right)\) be (a, b). Then the value of the integral $$\int\limits_a^b {{{{{\sin }^3}x} \over {({{\sin }^3}x + {{\sin }^3}(a +...

The value of the definite integral\(\int\limits_{ - {\pi \over 4}}^{{\pi \over 4}} {{{dx} \over {(1 + {e^{x\cos x}})({{\sin }^4}x + {{\cos }^4}x)}}}\) is equal to :

The value of \(\mathop {\lim }\limits_{n \to \infty } {1 \over n}\sum\limits_{j = 1}^n {{{(2j - 1) + 8n} \over {(2j - 1) + 4n}}}\) is equal to :

If \(\int_0^\pi {({{\sin }^3}x){e^{ - {{\sin }^2}x}}dx = \alpha - {\beta \over e}\int_0^1 {\sqrt t {e^t}dt} }\), then \(\alpha\) + \(\beta\) is equal to ____________.

Let f : (a, b) \(\to\) R be twice differentiable function such that \(f(x) = \int_a^x {g(t)dt}\) for a differentiable function g(x). If f(x) = 0 has exactly five distinct roots in (a, b), then g(x)g'(x) = 0 has at least :

\(\int\limits_6^{16} {{{{{\log }_e}{x^2}} \over {{{\log }_e}{x^2} + {{\log }_e}({x^2} - 44x + 484)}}dx}\) is equal to :

If \({U_n} = \left( {1 + {1 \over {{n^2}}}} \right)\left( {1 + {{{2^2}} \over {{n^2}}}} \right)^2.....\left( {1 + {{{n^2}} \over {{n^2}}}} \right)^n\), then \(\mathop {\lim }\limits_{n \to \infty } {({U_n})^{{{ - 4} \over {{n^2}}}}}\) is eq...

The value of the integral \(\int\limits_0^1 {{{\sqrt x dx} \over {(1 + x)(1 + 3x)(3 + x)}}}\) is :

The value of \(\sum\limits_{n = 1}^{100} {\int\limits_{n - 1}^n {{e^{x - [x]}}dx} }\), where [ x ] is the greatest integer \(\le\) x, is :

The value of \(\int\limits_{ - \pi /2}^{\pi /2} {{{{{\cos }^2}x} \over {1 + {3^x}}}} dx\) is :

The value of the integral \(\int\limits_0^\pi {|{{\sin }\,}2x|dx}\) is ___________.

If \({I_{m,n}} = \int\limits_0^1 {{x^{m - 1}}{{(1 - x)}^{n - 1}}dx}\), for m, \(n \ge 1\), and \(\int\limits_0^1 {{{{x^{m - 1}} + {x^{n - 1}}} \over {{{(1 + x)}^{m + 1}}}}} dx = \alpha {I_{m,n}}\alpha \in R\), then \(\alpha\) equals _____...