The value of \(\int _{ -\frac { \pi }{ 4 } }^{ \frac { \pi }{ 4 } }{ \left( \frac { { 2x }^{ 7 }-{ 3x }^{ 5 }+{ 7x }^{ 3 }-x+1 }{ { cos }^{ 2 }x } \right) dx } \) is 

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

The value of  \(\int _{ 0 }^{ \pi }{ { sin }^{ 4 }xdx } \) is

The volume of solid of revolution of the region bounded by y² = x(a − x) about x-axis is

The value of \(\int _{ 0 }^{ a }{ { (\sqrt { { a }^{ 2 }-{ x }^{ 2 } } ) }^{ 3 } } dx\) is

\(\text { The value of } \int_{0}^{\frac{2}{3}} \frac{d x}{\sqrt{4-9 x^{2}}} \text { is }\)

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

For any value of \(n \in \mathbb{Z}, \int_{0}^{\pi} e^{\cos ^{2} x} \cos ^{3}[(2 n+1) x] d x\) is

If \(\int _{ 0 }^{ 2a }{ f(x) } dx=2\int _{ 0 }^{ a }{ f(x) } \) then __________

The value of \(\int _{ -\pi }^{ \pi }{ { sin }^{ 3 }x \ { cos }^{ 3 }x \ } dx\) is __________

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

The ratio of the volumes generated by revolving the ellipse \(\frac { { x }^{ 2 } }{ 9 } +\frac { { y }^{ 2 } }{ 4 } \) = 1 about major and minor axes is __________

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

The volume generated by the curve y² = 16x from x = 2 to x = 3 rotating about x - axis ......... cu. units

The volume when  \(y=\sqrt { 3+{ x }^{ 2 } } \) from x = 0 to x = 4 is rotated about x-axis is .................

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

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

Evaluate the following integrals using properties of integration:
\(\int _{ -\frac { \pi }{ 4 } }^{ \frac { \pi }{ 4 } }{ { sin }^{ 2 }xdx } \)

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

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 \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { e }^{ 3x } } cosxdx\)

Evaluate \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { e }^{ -x } } \)

Find the slope of the tangent to the curve \(y=\int _{ 0 }^{ x }{ \frac { dt }{ 1+{ t }^{ 3 } } stx=1 } \) 

Find the area bounded by the curve y=sin2x between the ordinates x=0.x=π and x-axis.

Find the volume of the solid y=x³,x=0,y=1 is revolved about the y-axis.

Find an approximate value of \(\int _{ 1 }^{ 1.5 }{ xdx } \) by applying the left-end rule with the partition {1.1, 1.2, 1.3, 1.4, 1.5}.

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

Evaluate: \(\int _{ 0 }^{ \frac { 1 }{ \sqrt { 2 } } }{ \frac { { sin }^{ -1 }x }{ { (1-{ x }^{ 2 }) }^{ \frac { 3 }{ 2 } } } dx } \)

Show that \(\int ^\frac{2\pi}{0}_{0}\) g(cos x)dx = 2 \(\int ^{\pi}_{0}\) g(cosx)dx where g(cos x) is a function of cos x

If f (x) = f (a + x), then \(\int _{ 0 }^{ 2a }{ f(x)dx=2\int _{ 0 }^{ a }{ f(x)dx } } \)

Evaluate the following definite integrals:
\(\int _{ 0 }^{ 1 }{ \frac { 1-{ x }^{ 2 } }{ { (1+{ x }^{ 2 }) }^{ 2 } } } dx\)

Evaluate the following integrals using properties of integration:
\(\int _{ -5 }^{ 5 }{ xcos } \left( \frac { { e }^{ x }-1 }{ { e }^{ x }+1 } \right) dx\)

Evaluate \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { \sqrt { cot \ x } }{ \sqrt { cot \ x } +\sqrt { tan \ x } } dx } \)

Evaluate \(\int _{ 0 }^{ \pi }{ \sqrt { 1+4{ sin }^{ 2 }\frac { x }{ 2 } -4sin\frac { x }{ 2 } dx } } \) 

Find the area of the region bounded between the parabolas y²  = 4x and x² = 4y.

Find, by integration, the volume of the container which is in the shape of a right circular conical frustum.

Find the value of ‘c’ for which the area bounded by the curve y=8x²-x⁵,the lines x=1,x=c and x-axis \(\frac { 16 }{ 3 } \)

AOB is the positive quadrant of the ellipse \(\frac { { x }^{ 2 } }{ { a }^{ 2 } } +\cfrac { { y }^{ 2 } }{ { b }^{ 2 } } =1\) where OA=a and OB=b.Find the area between the arc AB and chord AB of the elipse.

Show that the ratio of the area under the curve y=sinx and y=sin2x between x=0 and \(x=\frac { \pi }{ 3 } \) and x- axis are as 2 : 3.

Find the area enclosed by the parabolas 5x²-y=0 and 2x²-y+9=0.