The Definite Integral

Evaluate the definite integral $\int_{0}^{1} \sqrt[m]{x^{n}}\, dx$, $n>0$.

Evaluate the definite integral $\int_{0}^{1} \frac{dx}{\sqrt[m]{x^{n}}}$, $m>n$.

Evaluate the definite integral $\int_{0}^{1} \frac{dx}{x^{2}+1}$.

Evaluate the definite integral $\int_{0}^{\infty } \frac{dx}{x^{2}+3}$.

Evaluate the definite integral $\int_{-2}^{-1} \frac{dx}{x}$.

Evaluate the definite integral $\int_{0}^{\pi /3} \tg x\, dx$.

Evaluate the definite integral $\int_{0}^{a} \frac{x\, dx}{\sqrt{a^{2}-x^{2}}}$.

Evaluate the definite integral $\int_{0}^{a} \frac{dx}{\sqrt{a^{2}-x^{2}}}$.

Evaluate the definite integral $\int_{0}^{\infty } \frac{dx}{x^{2}+a^{2}}$.

Evaluate the definite integral $\int_{-1}^{1} \frac{dx}{x^{2}}$.

Evaluate the definite integral $\int_{-1}^{1} \frac{dx}{x^{2}-1}$.

Prove the equality

Prove the equality

Evaluate the definite integral $\int_{0}^{1} \frac{x^{2}-1}{(x+2)^{4}}\, dx$.

Evaluate the definite integral $\int_{0}^{1} \frac{x^{3}}{(x^{3}+1)^{2}}\, dx$.

Evaluate the definite integral $\int_{0}^{\infty } \frac{x^{2}(1-x^{2})}{(1+x^{2})^{3}}\, dx$.

Evaluate the definite integral $\int_{1}^{\infty } \frac{dx}{x(1+x^{4})^{3}}$.

Evaluate the definite integral $\int_{1}^{\infty } \frac{\sqrt{1+x^{2}}}{x^{4}}\, dx$.

Evaluate the definite integral $\int_{0}^{2\pi } \frac{a+\cos x}{1+2a\cos x+a^{2}}\, dx$.

Evaluate the definite integral $\int_{0}^{1} e^{-x}\sin \pi x\, dx$.

Evaluate the definite integral $\int_{0}^{\pi } x^{2}\sin nx\, dx$, $n$ an integer.

Evaluate the definite integral $\int_{0}^{\pi /4} \sqrt{\tg x}\, dx$.

Evaluate the definite integral $\int_{-1}^{1} e^{k\arcsin x}\, dx$.

Evaluate the definite integral $\int_{0}^{\infty } \frac{dx}{\sqrt{e^{x}+a^{2}}}$.

Evaluate the definite integral $\int_{0}^{1} \frac{x^{3}\arcsin x}{\sqrt{1-x^{2}}}\, dx$.

Prove that the area bounded by the arc of the sine curve $y=\sin x$ and the $Ox$ axis between the points $(0,0)$ and $(\pi ,0)$ equals two square units.

Prove that the area of the curvilinear trapezoid bounded below by the segment of the $Ox$ axis between the points $(N,0)$ and $(2N,0)$, on the sides by verticals, and above by the hyperbola $y=\frac{1}{x}$, equals $\ln 2$.

Prove that the area bounded by the lemniscate of Bernoulli $r^{2}=a^{2}\cos 2\varphi$ equals $a^{2}$.

Prove that the area of the segment between the parabola $y=ax^{2}$ and the line $y=h$ equals $\frac{2}{3}$ of the product of the base and the height. (Archimedes.)

Prove that the volume of a paraboloid of revolution having the same base and height as a given cylinder equals half the volume of the cylinder. (Archimedes.)

Liquid spills over the edge of a vessel in the shape of a cylinder with radius $a$ and height $h$. Prove that at the moment when half of the bottom becomes exposed, the volume of the remaining liquid equals $\frac{2}{3}a^{2}h$. (Archimedes.)

Prove that the volume of the pitcher obtained by revolving the sine curve $y=r+b\sin \frac{2\pi x}{h}$ about the axis $Ox$ for $0<x<h$ equals $\pi r^{2}h+\frac{1}{2}\pi b^{2}h$.

A rod of length $l$ rotates about one of its ends, making $n$ revolutions per second. Determine the tension at the point of attachment, given that the weight per unit length of the rod equals $\sigma$, and the centrifugal force for a mass $m$ moving in a circle of radius $r$ with angular velocity $\omega$ equals $mr\omega^{2}$.

Under a load $f$, a wire of length $l$ with cross-sectional area $s$ and Young's modulus $E$ undergoes an elongation $\Delta l$ equal to $\frac{fl}{Es}$. Determine the elongation of such a wire hanging vertically under its own weight. The specific weight of the wire's material equals $\delta$.

Under a load of $1$ kg a wire stretches by $1$ cm. What work must be done to stretch it by $4$ cm?

Find the water pressure on a vertical wall of triangular shape. The height of the triangle is $h$, and the base $a$ lies on the surface of the liquid.

The same question for a wall in the shape of a semicircle whose diameter $2a$ lies on the surface of the water.

What work must be expended to pile up a conical heap of sand with radius $1{,}2$ m and height $1$ m, if the specific weight of the sand is $2$?

Find the position of the center of gravity of a homogeneous cone.

Find the center of gravity of a hemisphere of radius $R$.

A cylinder with radius $15$ cm and height $60$ cm is filled with air under a pressure of $1$ kg/cm${}^{2}$. What work must be done during isothermal compression of the gas to half its volume?

A point moves along the axis $Ox$, starting from the point $(1,0)$, so that its speed is numerically equal to its abscissa. Where will it be after $10$ seconds from the start of the motion?

Electric charges repel one another with a force $\frac{e_{1}e_{2}}{r^{2}}$, where $e_{1}$ and $e_{2}$ are the magnitudes of the charges and $r$ is the distance between them. Determine the work required to bring a charge $e_{2}=1$ from infinity to a distance $R$ from a charge $e_{1}$.

The Earth attracts a body of mass $m$ with a force $f=mg\frac{R^{2}}{r^{2}}$, where $R$ is the radius of the Earth and $r$ is the distance of the body from the Earth's center ($r \geqslant R$). Determine the work needed to remove the body to infinity from the Earth's surface.

By Joule's law, the amount of heat released by a direct current equals $Rci^{2}t$, where $c=0{,}24$ is a constant, $R$ is the resistance, $t$ is the number of seconds, and $i$ is the current. Find the heat released for an alternating current $i=a\cos bt$.

By Torricelli's law, the speed of the outflowing liquid equals $\sqrt{2gh}$, where $h$ is the depth of the opening below the surface of the liquid. Determine the time for the liquid to flow out of a cylinder of height $h$ and cross-sectional area $s$ through an opening of area $\sigma$ in the bottom.

The same question for a conical funnel with vertex at the bottom; $s$ is the area of the base, $h$ is the height of the funnel.

A disk of thickness $h$ and radius $r$ consists of a material with specific weight $\delta$ and makes $n$ revolutions per second. What work will it perform while braking to a stop?

On maps in the Mercator projection, the scale of the image increases from the equator to the pole in such a way that the distance between two meridians remains constant. The distance between meridians drawn $10°$ apart equals $a$. Find the distance between the equator and the parallel $\varphi °$.