Definite Integration

For x > 0, if \(f(x) = \int\limits_1^x {{{{{\log }_e}t} \over {(1 + t)}}dt}\), then \(f(e) + f\left( {{1 \over e}} \right)\) is equal to :

Let \(f(x) = \int\limits_0^x {{e^t}f(t)dt + {e^x}}\) be a differentiable function for all x\(\in\)R. Then f(x) equals :

The value of \(\mathop {\lim }\limits_{n \to \infty } {1 \over n}\sum\limits_{r = 0}^{2n - 1} {{{{n^2}} \over {{n^2} + 4{r^2}}}}\) is :

The value of \(\int\limits_{{{ - 1} \over {\sqrt 2 }}}^{{1 \over {\sqrt 2 }}} {{{\left( {{{\left( {{{x + 1} \over {x - 1}}} \right)}^2} + {{\left( {{{x - 1} \over {x + 1}}} \right)}^2} - 2} \right)}^{{1 \over 2}}}dx}\) is :

The value of \(\int\limits_{ - {\pi \over 2}}^{{\pi \over 2}} {\left( {{{1 + {{\sin }^2}x} \over {1 + {\pi ^{\sin x}}}}} \right)} \,dx\) is

If the value of the integral \(\int\limits_0^5 {{{x + [x]} \over {{e^{x - [x]}}}}dx = \alpha {e^{ - 1}} + \beta }\), where \(\alpha\), \(\beta\) \(\in\) R, 5\(\alpha\) + 6\(\beta\) = 0, and [x] denotes the greatest integer less than or equ...

Let \(f:[0,\infty ) \to [0,\infty )\) be defined as \(f(x) = \int_0^x {[y]dy}\)where [x] is the greatest integer less than or equal to x. Which of the following is true?

The value of the definite integral \(\int\limits_{\pi /24}^{5\pi /24} {{{dx} \over {1 + \root 3 \of {\tan 2x} }}}\) is :

The value of the integral \(\int\limits_{ - 1}^1 {\log \left( {x + \sqrt {{x^2} + 1} } \right)dx}\) is :

If \(f(x) = \left\{ {\matrix{ {\int\limits_0^x {\left( {5 + \left| {1 - t} \right|} \right)dt,} } & {x > 2} \cr {5x + 1,} & {x \le 2} \cr } } \right.\), then

The value of \(\int\limits_{ - 1}^1 {{x^2}{e^{[{x^3}]}}} dx\), where [ t ] denotes the greatest integer \(\le\) t, is :

If \({I_n} = \int\limits_{{\pi \over 4}}^{{\pi \over 2}} {{{\cot }^n}x\,dx}\), then :

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

The value of \(\int\limits_{ - 2}^2 {|3{x^2} - 3x - 6|dx}\) is ___________.

\(\mathop {\lim }\limits_{x \to 0} {{\int\limits_0^{{x^2}} {\left( {\sin \sqrt t } \right)dt} } \over {{x^3}}}\) is equal to :

If \(\int\limits_{ - a}^a {\left( {\left| x \right| + \left| {x - 2} \right|} \right)} dx = 22\), (a > 2) and [x] denotes the greatest integer \(\le\) x, then\(\int\limits_{ - a}^a {\left( {x + \left[ x \right]} \right)} dx\) is equal to ...

Let f be a twice differentiable function defined on R such that f(0) = 1, f'(0) = 2 and f'(x) \(\ne\) 0 for all x \(\in\) R. If \(\left| {\matrix{ {f(x)} & {f'(x)} \cr {f'(x)} & {f''(x)} \cr } } \right|\) = 0, for all x$$ \i...

Let f(x) be a differentiable function defined on [0, 2] such that f'(x) = f'(2 \(-\) x) for all x\(\in\) (0, 2), f(0) = 1 and f(2) = e2. Then the value of \(\int\limits_0^2 {f(x)} dx\) is :

The value of the integral, \(\int\limits_1^3 {[{x^2} - 2x - 2]dx}\), where [x] denotes the greatest integer less than or equal to x, is :

If \(\int\limits_0^{100\pi } {{{{{\sin }^2}x} \over {{e^{\left( {{x \over \pi } - \left[ {{x \over \pi }} \right]} \right)}}}}dx = {{\alpha {\pi ^3}} \over {1 + 4{\pi ^2}}},\alpha \in R}\) where [x] is the greatest integer less than or eq...

The value of the integral \(\int\limits_{ - 1}^1 {{{\log }_e}(\sqrt {1 - x} + \sqrt {1 + x} )dx}\) is equal to:

Let a be a positive real number such that \(\int_0^a {{e^{x - [x]}}} dx = 10e - 9\) where [ x ] is the greatest integer less than or equal to x. Then a is equal to:

Let \(g(t) = \int_{ - \pi /2}^{\pi /2} {\cos \left( {{\pi \over 4}t + f(x)} \right)} dx\), where \(f(x) = {\log _e}\left( {x + \sqrt {{x^2} + 1} } \right),x \in R\). Then which one of the following is correct?

If the real part of the complex number \({(1 - \cos \theta + 2i\sin \theta )^{ - 1}}\) is \({1 \over 5}\) for \(\theta \in (0,\pi )\), then the value of the integral \(\int_0^\theta {\sin x} dx\) is equal to:

If [x] denotes the greatest integer less than or equal to x, then the value of the integral \(\int_{ - \pi /2}^{\pi /2} {[[x] - \sin x]dx}\) is equal to :

The function f(x), that satisfies the condition \(f(x) = x + \int\limits_0^{\pi /2} {\sin x.\cos y\,f(y)\,dy}\), is :

Let \({J_{n,m}} = \int\limits_0^{{1 \over 2}} {{{{x^n}} \over {{x^m} - 1}}dx}\), \(\forall\) n > m and n, m \(\in\) N. Consider a matrix \(A = {[{a_{ij}}]_{3 \times 3}}\) where $${a_{ij}} = \left\{ {\matrix{
{{j_{6 + i,3}} - {j_{i + 3,3...

Let f : R \(\to\) R be a continuous function. Then \(\mathop {\lim }\limits_{x \to {\pi \over 4}} {{{\pi \over 4}\int\limits_2^{{{\sec }^2}x} {f(x)\,dx} } \over {{x^2} - {{{\pi ^2}} \over {16}}}}\) is equal to :

Let f(x) and g(x) be two functions satisfying f(x2) + g(4 \(-\) x) = 4x3 and g(4 \(-\) x) + g(x) = 0, then the value of \(\int\limits_{ - 4}^4 {f{{(x)}^2}dx}\) is

Let g(x) = \(\int_0^x {f(t)dt}\), where f is continuous function in [ 0, 3 ] such that \({1 \over 3}\) \(\le\) f(t) \(\le\) 1 for all t\(\in\) [0, 1] and 0 \(\le\) f(t) \(\le\) \({1 \over 2}\) for all t\(\in\) (1, 3]. The largest p...

Let P(x) be a real polynomial of degree 3 which vanishes at x = \(-\)3. Let P(x) have local minima at x = 1, local maxima at x = \(-\)1 and \(\int\limits_{ - 1}^1 {P(x)dx}\) = 18, then the sum of all the coefficients of the polynomial P(x)...

Which of the following statements is correct for the function g(\(\alpha\)) for \(\alpha\) \(\in\) R such that $$g(\alpha ) = \int\limits_{{\pi \over 6}}^{{\pi \over 3}} {{{{{\sin }^\alpha }x} \over {{{\cos }^\alpha }x + {{\sin }^\alpha }...

If [ . ] represents the greatest integer function, then the value of \(\left| {\int\limits_0^{\sqrt {{\pi \over 2}} } {\left[ {[{x^2}] - \cos x} \right]dx} } \right|\) is ____________.

If the integral
\(\int_0^{10} {{{[\sin 2\pi x]} \over {{e^{x - [x]}}}}} dx = \alpha {e^{ - 1}} + \beta {e^{ - {1 \over 2}}} + \gamma\), where \(\alpha\), \(\beta\), \(\gamma\) are integers and [x] denotes the greatest integer less than or...

Let \({I_n} = \int_1^e {{x^{19}}{{(\log |x|)}^n}} dx\), where n\(\in\)N. If (20)I10 = \(\alpha\)I9 + \(\beta\)I8, for natural numbers \(\alpha\) and \(\beta\), then \(\alpha\) \(-\) \(\beta\) equals to ___________.

Let f : R \(\to\) R be defined as f(x) = e\(-\)xsinx. If F : [0, 1] \(\to\) R is a differentiable function with that F(x) = \(\int_0^x {f(t)dt}\), then the value of \(\int_0^1 {(F'(x) + f(x)){e^x}dx}\) lies in the interval

If the normal to the curve y(x) = \(\int\limits_0^x {(2{t^2} - 15t + 10)dt}\) at a point (a, b) is parallel to the line x + 3y = \(-\)5, a > 1, then the value of | a + 6b | is equal to ___________.

Let f : (0, 2) \(\to\) R be defined as f(x) = log2\(\left( {1 + \tan \left( {{{\pi x} \over 4}} \right)} \right)\). Then, $$\mathop {\lim }\limits_{n \to \infty } {2 \over n}\left( {f\left( {{1 \over n}} \right) + f\left( {{2 \over n}} \r...

Let f : R \(\to\) R be a continuous function such that f(x) + f(x + 1) = 2, for all x\(\in\)R. If \({I_1} = \int\limits_0^8 {f(x)dx}\) and \({I_2} = \int\limits_{ - 1}^3 {f(x)dx}\), then the value of I1 + 2I2 is equal to ____________.

Let P(x) = x2 + bx + c be a quadratic polynomial with real coefficients such that \(\int_0^1 {P(x)dx}\) = 1 and P(x) leaves remainder 5 when it is divided by (x \(-\) 2). Then the value of 9(b + c) is equal to :

Consider the integral \(I = \int_0^{10} {{{[x]{e^{[x]}}} \over {{e^{x - 1}}}}dx}\), where [x] denotes the greatest integer less than or equal to x. Then the value of I is equal to :

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

If for all real triplets (a, b, c), ƒ(x) = a + bx + cx2;
then \(\int\limits_0^1 {f(x)dx}\) is equal to :

Let a function ƒ : [0, 5] \(\to\) R be continuous,
ƒ(1) = 3 and F be defined as :
\(F(x) = \int\limits_1^x {{t^2}g(t)dt}\) , where \(g(t) = \int\limits_1^t {f(u)du}\) Then for the function F, the point x = 1 is :

If \(I = \int\limits_1^2 {{{dx} \over {\sqrt {2{x^3} - 9{x^2} + 12x + 4} }}}\), then :

\(\mathop {\lim }\limits_{x \to 0} {{\int_0^x {t\sin \left( {10t} \right)dt} } \over x}\) is equal to

If ƒ(a + b + 1 - x) = ƒ(x), for all x, where a and b are fixed positive real numbers, then
\({1 \over {a + b}}\int_a^b {x\left( {f(x) + f(x + 1)} \right)} dx\) is equal to:

If \(\theta\)1
and \(\theta\)2
be respectively the smallest and the largest values of \(\theta\) in (0, 2\(\pi\)) - {\(\pi\)} which satisfy
the equation,
2cot2\(\theta\) - \({5 \over {\sin \theta }}\) + 4 = 0, then
$$\int\limits_{{\...

The value of \(\alpha\) for which
\(4\alpha \int\limits_{ - 1}^2 {{e^{ - \alpha \left| x \right|}}dx} = 5\), is:

\(\mathop {\lim }\limits_{x \to 1} \left( {{{\int\limits_0^{{{\left( {x - 1} \right)}^2}} {t\cos \left( {{t^2}} \right)dt} } \over {\left( {x - 1} \right)\sin \left( {x - 1} \right)}}} \right)\)