Definite Integration

The value of the definite integral\(\int\limits_{0}^{2} \frac{1}{3^x + 3} dx\)is

If
\(\alpha=\int\limits_{\frac{1}{2}}^2 \frac{\tan ^{-1} x}{2 x^2-3 x+2} d x\)
then the value of $\sqrt{7} \tan \left(\frac{2 \alpha \sqrt{7}}{\pi}\right)$ is _________.
(Here, the inverse trigonometric function $\tan ^{-1} x$ assumes val...

The value of $\frac{16}{\pi^3} \int\limits_0^{\frac{\pi}{2}} f(x) g(x) d x$ is ______.

The value of $2 \int\limits_0^{\frac{\pi}{2}} f(x) g(x) d x-\int\limits_0^{\frac{\pi}{2}} g(x) d x$ is ____________.

For $x \in \mathbb{R}$, let $\tan ^{-1}(x) \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$. Then the minimum value of the function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x)=\int\limits_0^{x \tan ^{-1} x} \frac{e^{(t-\cos t)}}{1...

Let $f:(0,1) \rightarrow \mathbb{R}$ be the function defined as $f(x)=\sqrt{n}$ if $x \in\left[\frac{1}{n+1}, \frac{1}{n}\right)$ where $n \in \mathbb{N}$. Let $g:(0,1) \rightarrow \mathbb{R}$ be a function such that $\int\limits_{x^2}^x \s...

The greatest integer less than or equal to

\(\int_{1}^{2} \log _{2}\left(x^{3}+1\right) d x+\int_{1}^{\log _{2} 9}\left(2^{x}-1\right)^{\frac{1}{3}} d x\)

is ___________.

Consider the equation
\(\int_{1}^{e} \frac{\left(\log _{\mathrm{e}} x\right)^{1 / 2}}{x\left(a-\left(\log _{\mathrm{e}} x\right)^{3 / 2}\right)^{2}} d x=1, \quad a \in(-\infty, 0) \cup(1, \infty)\)

Which of the following statements is/ar...

For any real number x, let [ x ] denote the largest integer less than or equal to x. If \(I = \int\limits_0^{10} {\left[ {\sqrt {{{10x} \over {x + 1}}} } \right]dx}\), then the value of 9I is __________.

Which of the following statements is TRUE?

Which of the following statements is TRUE?

Let \({g_i}:\left[ {{\pi \over 8},{{3\pi } \over 8}} \right] \to R,i = 1,2\), and \(f:\left[ {{\pi \over 8},{{3\pi } \over 8}} \right] \to R\) be functions such that \({g_1}(x) = 1,{g_2}(x) = |4x - \pi |\) and \(f(x) = {\sin ^2}x\), for a...

Let \({g_i}:\left[ {{\pi \over 8},{{3\pi } \over 8}} \right] \to R,i = 1,2\), and \(f:\left[ {{\pi \over 8},{{3\pi } \over 8}} \right] \to R\) be functions such that \({g_1}(x) = 1,{g_2}(x) = |4x - \pi |\) and \(f(x) = {\sin ^2}x\), for a...

Let \(f:\left[ { - {\pi \over 2},{\pi \over 2}} \right] \to R\) be a continuous function such that \(f(0) = 1\) and \(\int_0^{{\pi \over 3}} {f(t)dt = 0}\). Then which of the following statements is(are) TRUE?

Let b be a nonzero real number. Suppose f : R \(\to\) R is a differentiable function such that f(0) = 1. If the derivative f' of f satisfies the equation \(f'(x) = {{f(x)} \over {{b^2} + {x^2}}}\)for all x\(\in\)R, then which of the fol...

Let \(f:R \to R\) be a differentiable function such that its derivative f' is continuous and f(\(\pi\)) = \(-\)6.If \(F:[0,\pi ] \to R\) is defined by \(F(x) = \int_0^x {f(t)dt}\), and if \(\int_0^\pi {(f'(x)} + F(x))\cos x\,dx\) = 2the...

Which of the following inequalities is/are TRUE?

The value of the integral \(\int\limits_0^{\pi /2} {{{3\sqrt {\cos \theta } } \over {{{(\sqrt {\cos \theta } + \sqrt {\sin \theta } )}^5}}}} d\theta\) equals ..............

If \(I = {2 \over \pi }\int\limits_{ - \pi /4}^{\pi /4} {{{dx} \over {(1 + {e^{\sin x}})(2 - \cos 2x)}}}\), then 27I2 equals .................

The value of the integral\(\int_0^{1/2} {{{1 + \sqrt 3 } \over {{{({{(x + 1)}^2}{{(1 - x)}^6})}^{1/4}}}}dx}\) is ........

If \(I = \sum\nolimits_{k = 1}^{98} {\int_k^{k + 1} {{{k + 1} \over {x(x + 1)}}} dx}\), then

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

Let $$f\left( x \right) = \mathop {\lim }\limits_{n \to \infty } {\left( {{{{n^n}\left( {x + n} \right)\left( {x + {n \over 2}} \right)...\left( {x + {n \over n}} \right)} \over {n!\left( {{x^2} + {n^2}} \right)\left( {{x^2} + {{{n^2}} \ove...

The total number of distinct \(x \in \left[ {0,1} \right]\) for which \(\int\limits_0^x {{{{t^2}} \over {1 + {t^4}}}} dt = 2x - 1\)

Let \(f\left( x \right) = 7{\tan ^8}x + 7{\tan ^6}x - 3{\tan ^4}x - 3{\tan ^2}x\) for all \(x \in \left( { - {\pi \over 2},{\pi \over 2}} \right).\)
Then the correct expression(s) is (are)

The option(s) with the values of a and \(L\) that satisfy the following equation is (are)
$$${{\int\limits_0^{4\pi } {{e^t}\left( {{{\sin }^6}at + {{\cos }^4}at} \right)dt} } \over {\int\limits_0^\pi {{e^t}\left( {{{\sin }^6}at + {{\cos }...

If \(\alpha = \int\limits_0^1 {\left( {{e^{9x + 3{{\tan }^{ - 1}}x}}} \right)\left( {{{12 + 9{x^2}} \over {1 + {x^2}}}} \right)} dx\) where \({\tan ^{ - 1}}x\) takes only principal values, then the value of $$\left( {{{\log }_e}\left| {1 ...

Let \(f'\left( x \right) = {{192{x^3}} \over {2 + {{\sin }^4}\,\pi x}}\) for all \(x \in R\,\,\) with \(f\left( {{1 \over 2}} \right) = 0\).
If \(m \le \int\limits_{1/2}^1 {f\left( x \right)dx \le M,}\) then the possible values of \(m\) an...

Let \(f:R \to R\) be a function defined by \(f\left( x \right) = \left\{ {\matrix{ {\left[ x \right],} & {x \le 2} \cr {0,} & {x > 2} \cr } } \right.\) where \(\left[ x \right]\) is the greatest integer less than or equal to $$...

The following integral \(\int\limits_{{\pi \over 4}}^{{\pi \over 2}} {{{\left( {2\cos ec\,\,x} \right)}^{17}}dx}\) is equal to

List - \(I\)
P.\(\,\,\,\,\) The number of polynomials \(f(x)\) with non-negative integer coefficients of degree \(\le 2\), satisfying \(f(0)=0\) and \(\int_0^1 {f\left( x \right)dx = 1,}\) is
Q.\(\,\,\,\,\) The number of points in the in...

Given that for each \(a \in \left( {0,1} \right),\,\,\,\mathop {\lim }\limits_{h \to {0^ + }} \,\int\limits_h^{1 - h} {{t^{ - a}}{{\left( {1 - t} \right)}^{a - 1}}dt}\) exists. Let this limit be \(g(a).\) In addition, it is given that the ...

Given that for each \(a \in \left( {0,1} \right),\,\,\,\mathop {\lim }\limits_{h \to {0^ + }} \,\int\limits_h^{1 - h} {{t^{ - a}}{{\left( {1 - t} \right)}^{a - 1}}dt}\) exists. Let this limit be \(g(a).\) In addition, it is given that the ...

Let \(f:\left( {0,\infty } \right) \to R\) be given by \(f\left( x \right)\)= \(\int\limits_{{1 \over x}}^x {{{{e^{ - \left( {t + {1 \over t}} \right)}}} \over t}} dt\). Then

The value of \(\int\limits_0^1 {4{x^3}\left\{ {{{{d^2}} \over {d{x^2}}}{{\left( {1 - {x^2}} \right)}^5}} \right\}dx}\) is

Let a \(\in\) R and f : R \(\to\) R be given by f(x) = x5 \(-\) 5x + a. Then,

Let \(f\) \(:\,\,\left[ {{1 \over 2},1} \right] \to R\) (the set of all real number) be a positive,
non-constant and differentiable function such that
\(f'\left( x \right) < 2f\left( x \right)\) and \(f\left( {{1 \over 2}} \right) = 1.\) ...

The value of the integral \(\int\limits_{ - \pi /2}^{\pi /2} {\left( {{x^2} + 1n{{\pi + x} \over {\pi - x}}} \right)\cos xdx}\) is

The value of \(\,\int\limits_{\sqrt {\ell n2} }^{\sqrt {\ell n3} } {{{x\sin {x^2}} \over {\sin {x^2} + \sin \left( {\ell n6 - {x^2}} \right)}}\,dx}\) is

Let \(f\) be a real-valued function defined on the interval \((-1, 1)\) such that
\({e^{ - x}}f\left( x \right) = 2 + \int\limits_0^x {\sqrt {{t^4} + 1} \,\,dt,}\) for all \(x \in \left( { - 1,1} \right)\),
and let \({f^{ - 1}}\) be the ...

The value of \(\mathop {\lim }\limits_{x \to 0} {1 \over {{x^3}}}\int\limits_0^x {{{t\ln \left( {1 + t} \right)} \over {{t^4} + 4}}} dt\) is

The value of \(\int\limits_0^1 {{{{x^4}{{\left( {1 - x} \right)}^4}} \over {1 + {x^2}}}dx}\) is (are)

For any real number \(x,\) let \(\left[ x \right]\) denote the largest integer less than or equal to \(x.\) Let \(f\) be a real valued function defined on the interval \(\left[ { - 10,10} \right]\) by
$$$f\left( x \right) = \left\{ {\matrix...

Let \(f:R \to R\) be a continuous function which satisfies \(f(x) = \int\limits_0^x {f(t)dt}\). Then, the value of \(f(\ln 5)\) is ____________.

If \({I_n} = \int\limits_{ - \pi }^\pi {{{\sin nx} \over {(1 + {\pi ^x})\sin x}}dx,n = 0,1,2,}\) .... then

Let \(g\left( x \right) = \int\limits_0^{{e^x}} {{{f'\left( t \right)} \over {1 + {t^2}}}} \,dt.\) Which of the following is true?

\(\int\limits_{ - 1}^1 {g'\left( x \right)dx = }\)

Match the integrals in Column I with the values in Column II.

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\(\mathop {\lim }\limits_{x \to {\pi \over 4}} {{\int\limits_2^{{{\sec }^2}x} {f(t)\,dt} } \over {{x^2} - {{{\pi ^2}} \over {16}}}}\) equal

Match the integrals in Column \(I\) with the values in Column \(II\) and indicate your answer by darkening the appropriate bubbles in the \(4 \times 4\) matrix given in the \(ORS\).
Column \(I\)
(A) $$\int\limits_{ - 1}^1 {{{dx} \over {1 + ...