The value of \(\int _{ 0 }^{ \frac { \pi }{ 6 } }{ { cos }^{ 3 }3x\ dx }\ is\)

If \(f(x)=\int_{1}^{x} \frac{e^{\sin u}}{u} d u, x>1 \text { and }\int_{1}^{3} \frac{e^{\sin x^{2}}}{x} d x=\frac{1}{2}[f(a)-f(1)]\), then one of the possible value of a is

The value of \(\int _{ -\frac { \pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ { sin }^{ 2 }x\ cos \ x \ dx } \) is

The area enclosed by the curve y = \(\frac { { x }^{ 2 } }{ 2 } \) , the x - axis and the lines x = 1, x = 3 is __________

The area enclosed by the curve y² = 4x, the x-axis and its latus rectum is ________ sq.units.

The area of the region bounded by the graph of y = sin x and y = cos x between x = 0 and x = \(\frac { \pi }{ 4 } \)
(1) \(\int _{ 0 }^{ \frac { \pi }{ 4 } }{ (cos \ x-sin \ x) } dx\)
(2) \({ \left[ sinx+cosx \right] }_{ 0 }^{ \frac { \pi }{ 4 } }\)
(3) \(\int _{ 0 }^{ \frac { \pi }{ 4 } }{ (sinx-cosx) } dx\)
(4) \(\sqrt { 2 } -1\)

\(\int _{ a }^{ b }{ f(x) } dx=\)
(1) \(\int _{ a }^{ b }{ f(y) } dy\)
(2) \(-\int _{ a }^{ b }{ f(x) } dx\)
(3) \(\int _{ a }^{ b }{ f(a+b-x) } dx\)
(4) \(\int _{ 0 }^{ a }{ f(a-x) } dx\)​​

Evaluate:  \(\int ^{\frac{\pi}{2}}_{\frac{\pi}{2}}\)x cos x dx.

Evaluate the following definite integrals:
\(\int _{ 3 }^{ 4 }{ \frac { dx }{ { x }^{ 2 }-4 } } \)

Find the area of the region enclosed by the curve y = \(\sqrt x\) + 1, the axis of x and the lines x = 0, x = 4.

Evaluate \(\int _{ 1 }^{ 2 }{ \frac { x }{ (x+1)(x+2) } dx } \)

Evaluate the following definite integrals:
\(\int _{ -1 }^{ 1 }{ \frac { dx }{ { x }^{ 2 }+2x+5 } } \)

Evaluate \(\int _{ 0 }^{ \infty }{ \frac { { x }^{ n } }{ { x }^{ x } } } dx\), where n is positive integer \(\ge\)2

Find, by integration, the volume of the solid generated by revolving about y-axis the region bounded between the curve y =\(\frac{3}{4} \sqrt {x^2 -16}, x\ge4\) the y-axis, and the lines y = 1 and y = 6.

Evaluate \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { sin \ x }{ 9+{ cos }^{ 2 } } dx } \)

Show that \(\int ^\frac{\pi}{2}_0\) \(\frac {dx}{4+5 sin x}\) = \(\frac {1}{3}\) log_e 2.

Show that \(\int ^{1}_{0} (tan ^{-1} x + tan ^{-1}(1-x))\) dx = \(\frac {\pi}{2}\) - log_e2 

Show that the area under the curve y = sin x and y = sin 2x between x = 0 and x = \(\frac { \pi }{ 3 } \) and x axis are as 2:3