The value of \(\int _{ -4 }^{ 4 }{ \left[ { tan }^{ -1 }\left( \frac { { x }^{ 2 } }{ { x }^{ 4 }+1 } \right) +{ tan }^{ -1 }\left( \frac { { x }^{ 4 }+1 }{ { x }^{ 2 } } \right) \right] dx } \) is

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

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

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

\(\int _{ 1 }^{ \sqrt { 3 } }{ \frac { dx }{ 1+{ x }^{ 2 } } } \) is __________

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

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

The area bounded by the parabola y = x² and the line y = 2x is __________

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

\(\int _{ -1 }^{ 1 }{ x \ dx } \) = ...............

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

The area of the ellipse \(\frac { { x }^{ 2 } }{ 9 } +\frac { { y }^{ 2 } }{ 4 } =1\) __________

\(\int _{ -\frac { \pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ \frac { sinx }{ 2+cosx } dx= } \) __________

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

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

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

Evaluate \(\int _{ 0 }^{ 1 }{ \frac { { e }^{ x } }{ 1+{ e }^{ 2x } } dx } \)

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

Find the area of the region bounded by the curve y=sin x and the ordinate x=0.\(x=\frac { \pi }{ 3 } \)

Find the area bounded by the curve y=cosax in one arc of the curve.

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

Evaluate: \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { cos\theta }{ (1+sin\theta )(2+sin\theta ) } } d\theta \)

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

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

Evaluate the following integrals using properties of integration:
\(\int _{ 0 }^{ 2\pi }{ xlog\left( \frac { 3+cos\ x }{ 3-cos\ x } \right) } dx\)

Evaluate \(\\ \int _{ 0 }^{ 1 }{ { e }^{ -2x }(1+x-{ 2x }^{ 3 })dx } \)

Evaluate the following:
\(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { dx }{ 1+5{ cos }^{ 2 }x } } \)

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

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\(\int ^{\pi}_{0} \frac{x}{1+sin x}\) dx

Find the volume of the solid generated by the revolution of the loop of the curve x = t² y = t - \(\frac { { t }^{ 3 } }{ 3 } \) about x-axis.

Find the area of the curve y²=(x-5)²(x-6) between
(i) x=5 and x=6
(ii) x=6 and x=7

Find the area bounded by the curves y=|x|-1 and y=-|x|+1

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