Evaluate \(\int _{ 0 }^{ 1 }{ xdx } \), as the limit of a sum.

Evaluate \(\int _{ 0 }^{ 1 }{ x^3dx } \), as the limit of a sum.

Evaluate :\(\int _{ 0 }^{ 1 }{ [2x] } dx\) where [⋅] is the greatest integer function

Evaluate :\(\int _{ 0 }^{ \frac { \pi }{ 3 } }{ \frac { sec\ x\ tan\ x }{ 1+{ sec }^{ 2 }x } dx } \)

Evaluate :\(\int _{ 0 }^{ 9 }{ \frac { 1 }{ x+\sqrt { x } } dx } \)

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

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

Evaluate: \(\int ^{log 2}_{-log 2} e ^{-|x|}\) dx.

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

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

Evaluate \(\int _{ b }^{ \infty }{ \frac { 1 }{ { a }^{ 2 }+{ x }^{ 2 } } dx,a>0,b\in R } \)

Evaluate \(\int ^\frac {\pi}{2}_{0} \)( sin² x + cos⁴ x ) dx

Evaluate \(\int^\frac{\pi}{2}_0 \) \(\begin{vmatrix} { cos }^{ 4 }x & 7 \\ { sin }^{ 5 }x & 3 \end{vmatrix}\) dx

Find the values of the following:
\(\int ^\frac{\pi}{2}_{0}\)sin ⁵x cos⁴xdx

Find the values of the following:
\(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { sin }^{ 4 }x } x{ cos }^{ 6 }x\quad dx\)

Evaluate \(\int _{ 0 }^{ 1 }{ { x }^{ 3 }{ (1-x) }^{ 4 }dx } \)

Evaluate the following
\(\int _{ 0 }^{ \pi /2 }{ { sin }^{ 10 }x\quad dx } \)

Evaluate the following \(\int _{ 0 }^{ \pi /2 }{ { cos}^{ 7}x\quad dx } \)

Evaluate the following
\(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { sin }^{ 2 }x{ cos }^{ 4 }xdx } \)

Evaluate the following
\(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { sin }^{ 3 }\theta { cos }^{ 5 }\theta d\theta } \)

Evaluate \(\int _{ 0 }^{ \infty }{ { e }^{ -ax }{ x }^{ n }dx } \), where a > 0 .

Show that Γ(n) = 2\(\int _{ 0 }^{ \infty }{ { e }^{ -{ x }^{ 2 } }{ x }^{ 2n-1 }dx } \)

Find, by integration, the volume of the solid generated by revolving about y-axis the region bounded by the curves y = log x, y = 0, x = 0 and y = 2.