Definite Integration

\(f\left( x \right) = \left| {\matrix{ {\sec x} & {\cos x} & {{{\sec }^2}x + \cot x\cos ec\,x} \cr {{{\cos }^2}x} & {{{\cos }^2}x} & {\cos e{c^2}x} \cr 1 & {{{\cos }^2}x} & {{{\cos }^2}x} \cr } } \right|.\)
Then $$\int\limi...

Evaluate : \(\int\limits_0^\pi {{{x\,dx} \over {1 + \cos \,\alpha \,\sin x}},0 < \alpha < \pi }\)

For any integer \(n\) the integral ...........
\(\int\limits_0^\pi {{e^{{{\cos }^2}x}}{{\cos }^3}\left( {2n + 1} \right)xdx}\) has the value

Evaluate the following : \(\,\,\int\limits_0^{\pi /2} {{{x\sin x\cos x} \over {{{\cos }^4}x + {{\sin }^4}x}}} dx\)

Evaluate the following \(\int\limits_0^{{1 \over 2}} {{{x{{\sin }^{ - 1}}x} \over {\sqrt {1 - {x^2}} }}dx}\)

Given a function \(f(x)\) such that
(i) it is integrable over every interval on the real line and
(ii) \(f(t+x)=f(x),\) for every \(x\) and a real \(t\), then show that
the integral \(\int\limits_a^{a + 1} {f\,\,\left( x \right)} \,dx\) ...

The value of the integral \(\int\limits_0^{\pi /2} {{{\sqrt {\cot x} } \over {\sqrt {\cot x} + \sqrt {\tan x} }}dx}\) is

Evaluate : \(\int\limits_0^{\pi /4} {{{\sin x + \cos x} \over {9 + 16\sin 2x}}dx}\)

Find the value of \(\int\limits_{ - 1}^{3/2} {\left| {x\sin \,\pi \,x} \right|\,dx}\)

Show that \(\int\limits_0^\pi {xf\left( {\sin x} \right)dx} = {\pi \over 2}\int\limits_0^\pi {f\left( {\sin x} \right)dx.}\)

Show that : \(\mathop {\lim }\limits_{n \to \infty } \left( {{1 \over {n + 1}} + {1 \over {n + 2}} + .... + {1 \over {6n}}} \right) = \log 6\)

The value of the definite integral \(\int\limits_0^1 {\left( {1 + {e^{ - {x^2}}}} \right)} \,\,dx\)

Let \(a, b, c\) be non-zero real numbers such that
\(\int\limits_0^1 {\left( {1 + {{\cos }^8}x} \right)\left( {a{x^2} + bx + c} \right)dx = \int\limits_0^2 {\left( {1 + {{\cos }^8}x} \right)\left( {a{x^2} + bx + c} \right)dx.} }\)
Then th...