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}.

Find an approximate value of \(\int _{ 1 }^{ 1.5 }{ x^2dx } \) by applying the right-end rule with the partition {1.1, 1.2, 1.3, 1.4, 1.5}.

Find an approximate value of \(\int _{ 1 }^{ 1.5 }{ { (2-x)dx } } \) by applying the mid-point rule with the partition {1.1, 1.2, 1.3, 1.4, 1.5}.

Evaluate\(\int _{ 1 }^{ 4 }{ ({ 2x }^{ 2 }+3) } \) dx, as the limit of a sum

Evaluate the following integrals as the limits of sums.
\(\int _{ 0 }^{ 1 }{ (5x+4)dx } \)

Evaluate the following integrals as the limits of sums.
\(\int _{ 1 }^{ 2 }{( 4x^2-1)dx } \)

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

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

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

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

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

Evaluate: \(\int _{ 0 }^{ 1.5 }{ [{ x }^{ 2 }]dx } \) where [x] is the greatest integer function

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

Prove that \(\int ^\frac {\pi}{4}_{0} \frac{dx}{a^2 sin^2 x+b^2 cos^2 x}\) = \(\frac{1}{ab} tan^{-1} (\frac{a}{b})\) where a, b > 0

Show that \(\int _{ 0 }^{ \pi }{ g(sinx)dx=2 } \int _{ 0 }^{ \frac { \pi }{ 2 } }{ g(sinx)dx, } \) where g(sin x) is a function of sin x.

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

Evaluate the following definite integrals:
\(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { e }^{ x }\left( \frac { 1+sinx }{ 1+cosx } \right) dx } \)

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

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 the following integrals using properties of integration:
\(\int _{ -\frac { \pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ ({ x }^{ 5 }+xcos\ x+{ tan }^{ 3 }x+1)dx } \)

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

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

Evaluate the following integrals using properties of integration:
\(\int _{ 0 }^{ 2\pi }{ { sin }^{ 4 }{ x\ cos }^{ 3 }xdx } \)

Evaluate the following integrals using properties of integration:
\(\int _{ 0 }^{ 1 }{ |5x-3|dx } \)

Evaluate the following integrals using properties of integration:
\(\int _{ 0 }^{ { sin }^{ 2 }x }{ { sin }^{ -1 }\sqrt { t } dt+\int _{ 0 }^{ { cos }^{ 2 }x }{ { cos }^{ -1 }\sqrt { t } dt } } \)

Evaluate \(\int _{ 0 }^{ x }{ { x }^{ 2 } } \)cos nx dx, where n is a positive integer.

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

Evaluate: \(\int _{ 0 }^{ 2\pi }{ { x }^{ 2 }sin\ nx\ dx } \) where n is a positive integer.

Evaluate: \(\\ \\ \int _{ -1 }^{ 1 }{ { e }^{ -\lambda x }(1-{ x }^{ 2 }) } dx\)

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

Evaluate the following:
\(\int _{ 0 }^{ 1 }{ \frac { sin(3{ tan }^{ -1 }x){ tan }^{ -1 }x }{ 1+{ x }^{ 2 } } } dx\)

Evaluate the following:
\(\int _{ 0 }^{ \frac { 1 }{ 2 } }{ \frac { { e }^{ { a\ sin }^{ -1x } }{ sin }^{ -1 }x }{ \sqrt { 1-{ x }^{ 2 } } } dx } \)

Evaluate the following:
\(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { x }^{ 2 }cos2x\ dx } \)

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

Evaluate \(\int _{ 0 }^{ 2a }{ { x }^{ 2 }\sqrt { 2ax-{ x }^{ 2 } } } dx\)

Evaluate \(\int _{ 0 }^{ 1 }{ { x }^{ 5 }{ (1-{ x }^{ 2 }) }^{ 5 }dx } \)

Evaluate the following
\(\int _{ 0 }^{ \pi /4 }{ { sin}^{ 6}2x\ dx } \)

Evaluate the following
\(\int _{ 0 }^{ \pi /6 }{ { sin}^{ 5}3x\ dx } \)

Evaluate the following
\(\int _{ 0 }^{ 2\pi }{ { sin }^{ 7 } } \frac { x }{ 4 } dx\)

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

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

Evaluate the following:
\(\int _{ 0 }^{ \infty }{ { x }^{ 5 }{ e }^{ -3x }dx } \)

Evaluate the following:
\(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { { e }^{ -tanx } }{ { cos }^{ 6 }x } } dx\)

If \(\int _{ 0 }^{ \infty }{ { e }^{ -a{ x }^{ 2 } }{ x }^{ 3 }dx=32,\alpha >0,\ find\ \alpha } \)

Find the area of the region bounded by the line 6x + 5y = 30, x − axis and the lines x = −1 and x = 3.

Find the area of the region bounded by the y-axis and the parabola x = 5 − 4y − y².

Find the area of the region bounded by x−axis, the curve y = |cos x|, the lines x = 0 and x = \(\pi\).

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

Using integration find the area of the region bounded by triangle ABC, whose vertices A, B, and C are (−1, 1), (3, 2), and (0, 5) respectively

Using integration, find the area of the region which is bounded by x-axis, the tangent and normal to the circle x² +  y² = 4 drawn at (1, \(\sqrt 3\))

Find the volume of a sphere of radius a.

Find the volume of a right-circular cone of base radius r and height h.

Find the volume of the solid formed by revolving the region bounded by the parabola y = x², x-axis, ordinates x = 0 and x = 1 about the x-axis.

Find the volume of the solid formed by revolving the region bounded by the ellipse \(\frac { { x }^{ 2 } }{ { a }^{ 2 } } +\frac { { y }^{ 2 } }{ { b }^{ 2 } } =1\), a>b about the major axis.

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.

If f (x) be a continuous function defined on a closed interval [a,b] and\(F(x)=\int_{a}^{x} f(u) d u, \quad a then, \(\frac{d}{d x} F(x)=f(x) . \text { In other words, } F(x)\) is an anti-derivative of f (x).

If f (x) be a continuous function defined on a closed interval [a,b] and F(x) is an anti-derivative of f (x), then,
\(\int_{a}^{b} f(x) d x=F(b)-F(a)\)