Estimate the value of \(\int _{ 0 }^{ 0.5 }{ { x }^{ 2 } } dx\) using the Riemann sums corresponding to 5 subintervals of equal width and applying
(i) left-end rule
(ii) right-end rule
(iii) the mid-point rule.

Evaluate: \(\int _{ 0 }^{ 3 }{ (3{ x }^{ 2 }-4x+5) } dx\)

Evaluate :\(\int _{ 0 }^{ 1 }{ \frac { 2x+7 }{ { 5x }^{ 2 }+9 } } dx\)

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

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

Prove that \(\int^{\frac{\pi}{4}}_{0} \frac {sin 2x dx}{ sin ^4x +cos ^4 x}\) = \(\frac{\pi}{4}\) 

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

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

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

Evaluate \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { dx }{ 4{ sin }^{ 2 }x+5{ cos }^{ 2 }x } } \)

Prove that \(\int _{ 0 }^{ \infty }{ { e }^{ -x }{ x }^{ n }dx=n! } \) where n is a positive integer.

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 parabola x = y² +1, the y-axis, and the lines y = 1 and y = −1.

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.

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.