Definite Integration

Let \(f(x)=\frac{x}{\left(1+x^{n}\right)^{\frac{1}{n}}}, x \in \mathbb{R}-\{-1\}, n \in \mathbb{N}, n > 2\).
If \(f^{n}(x)=\left(f \circ f \circ f \ldots .\right.\). upto \(n\) times) \((x)\), then
$$\lim _\limits{n \rightarrow \infty} \in...

\(\lim _\limits{n \rightarrow \infty}\left\{\left(2^{\frac{1}{2}}-2^{\frac{1}{3}}\right)\left(2^{\frac{1}{2}}-2^{\frac{1}{5}}\right) \ldots . .\left(2^{\frac{1}{2}}-2^{\frac{1}{2 n+1}}\right)\right\}\) is equal to :

Let \(f(x)\) be a function satisfying \(f(x)+f(\pi-x)=\pi^{2}, \forall x \in \mathbb{R}\). Then \(\int_\limits{0}^{\pi} f(x) \sin x d x\) is equal to :

The value of \(\int_\limits{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{(2+3 \sin x)}{\sin x(1+\cos x)} d x\) is equal to :

Let \(\alpha \in (0,1)\) and \(\beta = {\log _e}(1 - \alpha )\). Let \({P_n}(x) = x + {{{x^2}} \over 2} + {{{x^3}} \over 3}\, + \,...\, + \,{{{x^n}} \over n},x \in (0,1)\). Then the integral $$\int\limits_0^\alpha {{{{t^{50}}} \over {1 -...

If $\phi(x)=\frac{1}{\sqrt{x}} \int\limits_{\frac{\pi}{4}}^x\left(4 \sqrt{2} \sin t-3 \phi^{\prime}(t)\right) d t, x>0$,
then $\emptyset^{\prime}\left(\frac{\pi}{4}\right)$ is equal to :

Let $\alpha>0$. If $\int\limits_0^\alpha \frac{x}{\sqrt{x+\alpha}-\sqrt{x}} \mathrm{~d} x=\frac{16+20 \sqrt{2}}{15}$, then $\alpha$ is equal to :

\(\lim_\limits{x \rightarrow 0} \frac{48}{x^{4}} \int_\limits{0}^{x} \frac{t^{3}}{t^{6}+1} \mathrm{~d} t\) is equal to ___________.

If [t] denotes the greatest integer \(\le \mathrm{t}\), then the value of \({{3(e - 1)} \over e}\int\limits_1^2 {{x^2}{e^{[x] + [{x^3}]}}dx}\) is :

$\lim\limits_{n \rightarrow \infty} \frac{3}{n}\left\{4+\left(2+\frac{1}{n}\right)^2+\left(2+\frac{2}{n}\right)^2+\ldots+\left(3-\frac{1}{n}\right)^2\right\}$ is equal to :

Let \(f(x) = x + {a \over {{\pi ^2} - 4}}\sin x + {b \over {{\pi ^2} - 4}}\cos x,x \in R\) be a function which satisfies \(f(x) = x + \int\limits_0^{\pi /2} {\sin (x + y)f(y)dy}\). then \((a+b)\) is equal to

The value of the integral \(\int\limits_{1/2}^2 {{{{{\tan }^{ - 1}}x} \over x}dx}\) is equal to :

The value of the integral \(\int_1^2 {\left( {{{{t^4} + 1} \over {{t^6} + 1}}} \right)dt}\) is

The minimum value of the function \(f(x) = \int\limits_0^2 {{e^{|x - t|}}dt}\) is :

If \(\int\limits_{{1 \over 3}}^3 {|{{\log }_e}x|dx = {m \over n}{{\log }_e}\left( {{{{n^2}} \over e}} \right)}\), where m and n are coprime natural numbers, then \({m^2} + {n^2} - 5\) is equal to _____________.

The integral \(16\int\limits_1^2 {{{dx} \over {{x^3}{{\left( {{x^2} + 2} \right)}^2}}}}\) is equal to

The value of \({8 \over \pi }\int\limits_0^{{\pi \over 2}} {{{{{(\cos x)}^{2023}}} \over {{{(\sin x)}^{2023}} + {{(\cos x)}^{2023}}}}dx}\) is ___________

The value of \(12\int\limits_0^3 {\left| {{x^2} - 3x + 2} \right|dx}\) is ____________

Let \(f\) be \(a\) differentiable function defined on \(\left[ {0,{\pi \over 2}} \right]\) such that \(f(x) > 0\) and \(f(x) + \int_0^x {f(t)\sqrt {1 - {{({{\log }_e}f(t))}^2}} dt = e,\forall x \in \left[ {0,{\pi \over 2}} \right]}\). The...

\(\int\limits_{{{3\sqrt 2 } \over 4}}^{{{3\sqrt 3 } \over 4}} {{{48} \over {\sqrt {9 - 4{x^2}} }}dx}\) is equal to :

Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a differentiable function such that \(f^{\prime}(x)+f(x)=\int_\limits{0}^{2} f(t) d t\). If \(f(0)=e^{-2}\), then \(2 f(0)-f(2)\) is equal to ____________.

If \(\int_\limits{0}^{1}\left(x^{21}+x^{14}+x^{7}\right)\left(2 x^{14}+3 x^{7}+6\right)^{1 / 7} d x=\frac{1}{l}(11)^{m / n}\) where \(l, m, n \in \mathbb{N}, m\) and \(n\) are coprime then \(l+m+n\) is equal to ____________.

\(\mathop {\lim }\limits_{n \to \infty } \left[ {{1 \over {1 + n}} + {1 \over {2 + n}} + {1 \over {3 + n}}\, + \,...\, + \,{1 \over {2n}}} \right]\) is equal to

If \(\int\limits_0^\pi {{{{5^{\cos x}}(1 + \cos x\cos 3x + {{\cos }^2}x + {{\cos }^3}x\cos 3x)dx} \over {1 + {5^{\cos x}}}} = {{k\pi } \over {16}}}\), then k is equal to _____________.

The value of the integral \(\int\limits_{ - {\pi \over 4}}^{{\pi \over 4}} {{{x + {\pi \over 4}} \over {2 - \cos 2x}}dx}\) is :

If $\int\limits_{0}^{1} \frac{1}{\left(5+2 x-2 x^{2}\right)\left(1+e^{(2-4 x)}\right)} d x=\frac{1}{\alpha} \log _{e}\left(\frac{\alpha+1}{\beta}\right), \alpha, \beta>0$, then $\alpha^{4}-\beta^{4}$

is equal to :

Let for \(x \in \mathbb{R}, S_{0}(x)=x, S_{k}(x)=C_{k} x+k \int_{0}^{x} S_{k-1}(t) d t\), where
\(C_{0}=1, C_{k}=1-\int_{0}^{1} S_{k-1}(x) d x, k=1,2,3, \ldots\) Then \(S_{2}(3)+6 C_{3}\) is equal to ____________.

\(\int_\limits{0}^{\infty} \frac{6}{e^{3 x}+6 e^{2 x}+11 e^{x}+6} d x=\)

Among
(S1): \(\lim_\limits{n \rightarrow \infty} \frac{1}{n^{2}}(2+4+6+\ldots \ldots+2 n)=1\)
(S2) : \(\lim_\limits{n \rightarrow \infty} \frac{1}{n^{16}}\left(1^{15}+2^{15}+3^{15}+\ldots \ldots+n^{15}\right)=\frac{1}{16}\)

Let \(f_{n}=\int_\limits{0}^{\frac{\pi}{2}}\left(\sum_\limits{k=1}^{n} \sin ^{k-1} x\right)\left(\sum_\limits{k=1}^{n}(2 k-1) \sin ^{k-1} x\right) \cos x d x, n \in \mathbb{N}\). Then \(f_{21}-f_{20}\) is equal to _________

The value of \({{{e^{ - {\pi \over 4}}} + \int\limits_0^{{\pi \over 4}} {{e^{ - x}}{{\tan }^{50}}xdx} } \over {\int\limits_0^{{\pi \over 4}} {{e^{ - x}}({{\tan }^{49}}x + {{\tan }^{51}}x)dx} }}\) is

If \(\int_\limits{-0.15}^{0.15}\left|100 x^{2}-1\right| d x=\frac{k}{3000}\), then \(k\) is equal to ___________.

For \(m, n > 0\), let \(\alpha(m, n)=\int_\limits{0}^{2} t^{m}(1+3 t)^{n} d t\). If \(11 \alpha(10,6)+18 \alpha(11,5)=p(14)^{6}\), then \(p\) is equal to ___________.

The value of the integral \(\int_\limits{-\log _{e} 2}^{\log _{e} 2} e^{x}\left(\log _{e}\left(e^{x}+\sqrt{1+e^{2 x}}\right)\right) d x\) is equal to :

Let the function \(f:[0,2] \rightarrow \mathbb{R}\) be defined as
$$f(x)= \begin{cases}e^{\min \left\{x^{2}, x-[x]\right\},} & x \in[0,1) \\ e^{\left[x-\log _{e} x\right]}, & x \in[1,2]\end{cases}$$
where \([t]\) denotes the greatest intege...

If \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a continuous function satisfying \(\int_\limits{0}^{\frac{\pi}{2}} f(\sin 2 x) \sin x d x+\alpha \int_\limits{0}^{\frac{\pi}{4}} f(\cos 2 x) \cos x d x=0\), then the value of \(\alpha\) is :

Let \(f\) be a continuous function satisfying \(\int_\limits{0}^{t^{2}}\left(f(x)+x^{2}\right) d x=\frac{4}{3} t^{3}, \forall t > 0\). Then

\(f\left(\frac{\pi^{2}}{4}\right)\) is equal to :

Let \(f(t) = \int\limits_0^t {{e^{{x^3}}}\left( {{{{x^8}} \over {{{({x^6} + 2{x^3} + 2)}^2}}}} \right)dx}\). If \(f(1) + f'(1) = \alpha e - {1 \over 6}\), then the value of 150\(\alpha\) is equal to ___________.

\(\mathop {\lim }\limits_{n \to \infty } \sum\limits_{r = 1}^n {{r \over {2{r^2} - 7rn + 6{n^2}}}}\) is equal to :

\(\int_0^5 {\cos \left( {\pi \left( {x - \left[ {{x \over 2}} \right]} \right)} \right)dx}\),
where [t] denotes greatest integer less than or equal to t, is equal to:

Let \(f:R \to R\) be a function defined by :
$$f(x) = \left\{ {\matrix{
{\max \,\{ {t^3} - 3t\} \,t \le x} & ; & {x \le 2} \cr
{{x^2} + 2x - 6} & ; & {2 < x < 3} \cr
{[x - 3] + 9} & ; & {3 \le x \le 5} \cr
{2x + 1} & ; & {...

If \(\int\limits_0^2 {\left( {\sqrt {2x} - \sqrt {2x - {x^2}} } \right)dx = \int\limits_0^1 {\left( {1 - \sqrt {1 - {y^2}} - {{{y^2}} \over 2}} \right)dy + \int\limits_1^2 {\left( {2 - {{{y^2}} \over 2}} \right)dy + I} } }\), then I equa...

Let f be a real valued continuous function on [0, 1] and \(f(x) = x + \int\limits_0^1 {(x - t)f(t)dt}\).
Then, which of the following points (x, y) lies on the curve y = f(x) ?

If \(f(\alpha)=\int\limits_{1}^{\alpha} \frac{\log _{10} \mathrm{t}}{1+\mathrm{t}} \mathrm{dt}, \alpha>0\), then \(f\left(\mathrm{e}^{3}\right)+f\left(\mathrm{e}^{-3}\right)\) is equal to :

The integral \(\int\limits_{0}^{\frac{\pi}{2}} \frac{1}{3+2 \sin x+\cos x} \mathrm{~d} x\) is equal to :

If \([t]\) denotes the greatest integer \(\leq t\), then the value of \(\int_{0}^{1}\left[2 x-\left|3 x^{2}-5 x+2\right|+1\right] \mathrm{d} x\) is :

Let [t] denote the greatest integer less than or equal to t. Then, the value of the integral \(\int\limits_0^1 {[ - 8{x^2} + 6x - 1]dx}\) is equal to :

Let f : R \(\to\) R be a continuous function satisfying f(x) + f(x + k) = n, for all x \(\in\) R where k > 0 and n is a positive integer. If \({I_1} = \int\limits_0^{4nk} {f(x)dx}\) and \({I_2} = \int\limits_{ - k}^{3k} {f(x)dx}\), then ...

Let f : R \(\to\) R be a differentiable function such that \(f\left( {{\pi \over 4}} \right) = \sqrt 2 ,\,f\left( {{\pi \over 2}} \right) = 0\) and \(f'\left( {{\pi \over 2}} \right) = 1\) and let $$g(x) = \int_x^{\pi /4} {(f'(t)\sec t +...

If \(\int\limits_{0}^{\sqrt{3}} \frac{15 x^{3}}{\sqrt{1+x^{2}+\sqrt{\left(1+x^{2}\right)^{3}}}} \mathrm{~d} x=\alpha \sqrt{2}+\beta \sqrt{3}\), where \(\alpha, \beta\) are integers, then \(\alpha+\beta\) is equal to __________.