Definite Integration

The value of \(\mathop {\lim }\limits_{n \to \infty } {1 \over n}\left\{ {{{\sec }^2}{\pi \over {4n}} + {{\sec }^2}{{2\pi } \over {4n}} + ... + {{\sec }^2}{{n\pi } \over {4n}}} \right\}\) is

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

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

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

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

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

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 \(\in\) N \(-\) {1}. Then I1 / I2 is equal to

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, \(I = \int_0^a {{{dx} \over {1 + f(x)}}}\) is equal to

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

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

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

If \(\phi (t) = \left\{ \matrix{ 1,\,for\,0 \le t < 1, \hfill \cr 0,\,otherwise \hfill \cr} \right.\), then \(\int\limits_{ - 300}^{3000} {\left( {\sum\limits_{r' = 2014}^{2016} {\phi (t - r')\phi (t - 2016)} } \right)} \,dt\) is

\(\int\limits_0^1 {\log \left( {{1 \over x} - 1} \right)} dx\) is equal to

If [x] denotes the greatest integer less than or equal to x, then the value of the integral \(\int\limits_0^2 {{x^2}[x]\,dx}\) equals

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The value of the integral \(\int\limits_{ - a}^a {{{x{e^{{x^2}}}} \over {1 + {x^2}}}dx}\) 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 \(\int\limits_0^\pi {|\cos x|dx}\) is

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

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

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

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