The Definite Integral

Divide the interval $[-1,4]$ into $n$ equal subintervals, and take the coordinates of the midpoints of these subintervals as the values of the independent variable $\xi_{i}$ ($i=0,1,\ldots ,n-1$). Find the integral sum $S_{n}$ of the function $f(x)=1+x$ over this interval.

Let

Divide the closed interval $[1,2]$ into $n$ parts such that the abscissas of the dividing points form a geometric series, to find the lower integral sum of the function $f(x)=x^{4}$ on $[1,2]$. What does the limit of this sum equal as $n\to \infty$?

Starting from the definition of the integral, find

where $v_{0}$ and $g$ are constants.

Using a suitable method of subdividing the interval of integration, regard the integral as the limit of the corresponding integral sum to compute the definite integral: $\displaystyle \int_{-1}^{2}x^{2}\, dx$.

Using a suitable method of subdividing the interval of integration, regard the integral as the limit of the corresponding integral sum to compute the definite integral: $\displaystyle \int_{0}^{1}a^{x}\, dx$ ($a>0$).

Using a suitable method of subdividing the interval of integration, regard the integral as the limit of the corresponding integral sum to compute the definite integral: $\displaystyle \int_{0}^{\frac{\pi }{2}}\sin x\, dx$.

Using a suitable method of subdividing the interval of integration, regard the integral as the limit of the corresponding integral sum to compute the definite integral: $\displaystyle \int_{0}^{x}\cos t\, dt$.

Using a suitable method of subdividing the interval of integration, regard the integral as the limit of the corresponding integral sum to compute the definite integral: $\displaystyle \int_{a}^{b}\dfrac {dx}{x^{2}}$ ($0<a<b$).

Using a suitable method of subdividing the interval of integration, regard the integral as the limit of the corresponding integral sum to compute the definite integral: $\displaystyle \int_{a}^{b}x^{m}\, dx$ ($0<a<b$; $m\neq -1$).

Using a suitable method of subdividing the interval of integration, regard the integral as the limit of the corresponding integral sum to compute the definite integral: $\displaystyle \int_{a}^{b}\dfrac {dx}{x}$ ($0<a<b$).

Using a suitable method of subdividing the interval of integration, regard the integral as the limit of the corresponding integral sum, compute the Poisson integral

when:

Suppose the functions $f(x)$ and $\varphi (x)$ are continuous on $[a,b]$, prove that

where $x_{i}\leqslant \xi_{i}\leqslant x_{i+1}$, $x_{i}\leqslant \theta_{i}\leqslant x_{i+1}$ ($i=0,1,\ldots ,n-1$) and $\Delta x_{i}=x_{i+1}-x_{i}$ ($x_{0}=a$, $x_{n}=b$).

Prove that the discontinuous function:

is integrable on the interval $[0,1]$.

Prove that the Riemann function

(where $m$ and $n$ ($n\geqslant 1$) are coprime integers) is integrable on any finite interval.

Prove that the function

is integrable on the closed interval $[0,1]$.

Prove that the Dirichlet function

is not integrable on any interval.

Suppose the function $f(x)$ is integrable on $[a,b]$, and

where $x_{i}=a+\dfrac {i}{n}(b-a)$ ($i=0,1,\ldots ,n-1$; $n=1,2,\ldots$).

Prove that $\displaystyle \lim_{n\to \infty }\int_{a}^{b}f_{n}(x)\, dx=\int_{a}^{b}f(x)\, dx$.

Prove that if the function $f(x)$ is integrable on $[a,b]$, then there is a sequence of continuous functions $\varphi_{n}(x)$ ($n=1,2,\ldots$) such that

Prove that if the bounded function $f(x)$ is integrable on the closed interval $[a,b]$, then its absolute value $\left|f(x)\right|$ is also integrable on $[a,b]$, and

If the function $f(x)$ is absolutely integrable on the closed interval $[a,b]$ (that is, the integral $\displaystyle \int_{a}^{b}\left|f(x)\right|\, dx$ exists), is this function necessarily integrable on $[a,b]$?

Let the function $\varphi (x)$ be defined and continuous on the closed interval $[A,B]$, let the function $f(x)$ be integrable on $[a,b]$, and let $A\leqslant f(x)\leqslant B$ when $a\leqslant x\leqslant b$. Prove that the function $\varphi [f(x)]$ is integrable on $[a,b]$.

If the functions $f(x)$ and $\varphi (x)$ are integrable, must the function $f[\varphi (x)]$ also be integrable?

Let the function $f(x)$ be integrable on the closed interval $[A,B]$. Prove that the function $f(x)$ has continuity in the mean of the integral, that is,

where $[a,b]\subset [A,B]$.

Let the function $f(x)$ be integrable on the closed interval $[a,b]$. Prove that the equality

holds if and only if $f(x)=0$ at every point of the closed interval $[a,b]$ where $f(x)$ is continuous.

Using the Newton-Leibniz formula, find the following definite integral and sketch the corresponding curvilinear trapezoid area: $\displaystyle \int_{-1}^{8}\sqrt[3]{x} dx$.

Using the Newton-Leibniz formula, find the following definite integral and sketch the corresponding curvilinear trapezoid area: $\displaystyle \int_{0}^{\pi }\sin x dx$.

Using the Newton-Leibniz formula, find the following definite integral and sketch the corresponding curvilinear trapezoid area: $\displaystyle \int_{\frac{1}{\sqrt{3}}}^{\sqrt{3}}\dfrac {dx}{1+x^{2}}$.

Using the Newton-Leibniz formula, find the following definite integral and sketch the corresponding curvilinear trapezoid area: $\displaystyle \int_{-\frac{1}{2}}^{\frac{1}{2}}\dfrac {dx}{\sqrt{1-x^{2}}}$.

Using the Newton-Leibniz formula, find the following definite integral and sketch the corresponding curvilinear trapezoid area: $\displaystyle \int_{\operatorname {sh}1}^{\operatorname {sh}2}\dfrac {dx}{\sqrt{1+x^{2}}}$.

Using the Newton-Leibniz formula, find the following definite integral and sketch the corresponding curvilinear trapezoid area: $\displaystyle \int_{0}^{2}\left|1-x\right| dx$.

Using the Newton-Leibniz formula, find the following definite integral and sketch the corresponding curvilinear trapezoid area: $\displaystyle \int_{-1}^{1}\dfrac {dx}{x^{2}-2x\cos \alpha +1}$ ($0<\alpha <\pi$).

Using the Newton-Leibniz formula, find the following definite integral: $\displaystyle \int_{0}^{2\pi }\dfrac {dx}{1+\varepsilon \cos x}$ ($0\leqslant \varepsilon <1$).

Using the Newton-Leibniz formula, find the following definite integral: $\displaystyle \int_{-1}^{1}\dfrac {dx}{\sqrt{(1-2ax+a^{2})(1-2bx+b^{2})}}$ ($\left|a\right|<1$, $\left|b\right|<1$, $ab>0$).

Using the Newton-Leibniz formula, find the following definite integral: $\displaystyle \int_{0}^{\frac{\pi }{2}}\dfrac {dx}{a^{2}\sin^{2}x+b^{2}\cos^{2}x}$ ($ab\neq 0$).

Using the Newton-Leibniz formula, evaluate the following definite integrals, and explain why a formal application of the Newton-Leibniz formula yields an incorrect result:

Find $\displaystyle \int_{-1}^{1}\dfrac {d}{dx}\left(\dfrac {1}{1+2^{\frac{1}{x}}}\right) dx$.

Find $\displaystyle \int_{0}^{100\pi }\sqrt{1-\cos 2x} dx$.

Use a definite integral to find the limit of the following sum: $\displaystyle \lim_{n\to \infty }\left(\dfrac {1}{n^{2}}+\dfrac {2}{n^{2}}+\cdots +\dfrac {n-1}{n^{2}}\right)$.

Use a definite integral to find the limit of the following sum: $\displaystyle \lim_{n\to \infty }\left(\dfrac {1}{n+1}+\dfrac {1}{n+2}+\cdots +\dfrac {1}{n+n}\right)$.

Use a definite integral to find the limit of the following sum: $\displaystyle \lim_{n\to \infty }\left(\dfrac {n}{n^{2}+1^{2}}+\dfrac {n}{n^{2}+2^{2}}+\cdots +\dfrac {n}{n^{2}+n^{2}}\right)$.

Use a definite integral to find the limit of the following sum: $\displaystyle \lim_{n\to \infty }\dfrac {1}{n}\left[\sin \dfrac {\pi }{n}+\sin \dfrac {2\pi }{n}+\cdots +\sin \dfrac {(n-1)\pi }{n}\right]$.

Use a definite integral to find the limit of the following sum: $\displaystyle \lim_{n\to \infty }\dfrac {1^{p}+2^{p}+\cdots +n^{p}}{n^{p+1}}$ ($p>0$).

Use a definite integral to find the limit of the following sum: $\displaystyle \lim_{n\to \infty }\dfrac {1}{n}\left(\sqrt{1+\dfrac {1}{n}}+\sqrt{1+\dfrac {2}{n}}+\cdots +\sqrt{1+\dfrac {n}{n}}\right)$.

Use a definite integral to find the following limit: $\displaystyle \lim_{n\to \infty }\dfrac {\sqrt[n]{n!}}{n}$.

Use a definite integral to find the following limit: $\displaystyle \lim_{n\to \infty }\left[\dfrac {1}{n}\sum_{k=1}^{n}f\left(a+k\cdot \dfrac {b-a}{n}\right)\right]$.

Discarding higher-order infinitesimals of the same order, find the limit of the following sum: $\displaystyle \lim_{n\to \infty }\left[\left(1+\dfrac {1}{n}\right)\sin \dfrac {\pi }{n^{2}}+\left(1+\dfrac {2}{n}\right)\sin \dfrac {2\pi }{n^{2}}+\cdots +\left(1+\dfrac {n-1}{n}\right)\sin \dfrac {(n-1)\pi }{n^{2}}\right]$.

Use a definite integral to find the following limit: $\displaystyle \lim_{n\to \infty }\sin \dfrac {\pi }{n}\cdot \sum_{k=1}^{n}\dfrac {1}{2+\cos \frac{k\pi }{n}}$.

Use a definite integral to find the following limit: $\displaystyle \lim_{n\to \infty }\dfrac {\displaystyle \sum_{k=1}^{n}\sqrt{(nx+k)(nx+k+1)}}{n^{2}}$ ($x>0$).

Use a definite integral to find the following limit: $\displaystyle \lim_{n\to \infty }\left(\dfrac {2^{\frac{1}{n}}}{n+1}+\dfrac {2^{\frac{2}{n}}}{n+\frac{1}{2}}+\cdots +\dfrac {2^{\frac{n}{n}}}{n+\frac{1}{n}}\right)$.

Find: $\displaystyle \dfrac {d}{dx}\int_{a}^{b}\sin x^{2} dx$, $\displaystyle \dfrac {d}{da}\int_{a}^{b}\sin x^{2} dx$, $\displaystyle \dfrac {d}{db}\int_{a}^{b}\sin x^{2} dx$.

Find:

Find:

$\displaystyle \lim_{x\to 0}\dfrac {\int_{0}^{x}\cos x^{2} dx}{x}$;

$\displaystyle \lim_{x\to 0}\dfrac {\int_{0}^{x}\cos x^{2} dx}{x}$;

Prove that as $x\to \infty$, $\displaystyle \int_{0}^{x}e^{x^{2}} dx\sim \dfrac {1}{2x}e^{x^{2}}$.

Find: $\displaystyle \lim_{x\to +0}\dfrac {\int_{0}^{\sin x}\sqrt{\tan t} dt}{\int_{0}^{\tan x}\sqrt{\sin t} dt}$.

Let $f(x)$ be a continuous positive-valued function; prove that for $x\geqslant 0$, the function

is increasing.

Find:

Evaluate the following integrals and, treating them as functions of the parameter $\alpha$, sketch the graph of the integral $I=I(\alpha )$, where:

Using the integration-by-parts formula, find the following definite integral: $\displaystyle \int_{0}^{\ln 2}xe^{-x} dx$.

Using the integration-by-parts formula, find the following definite integral: $\displaystyle \int_{0}^{\pi }x\sin x dx$.

Using the integration-by-parts formula, find the following definite integral: $\displaystyle \int_{0}^{2\pi }x^{2}\cos x dx$.

Using the integration-by-parts formula, evaluate the following definite integral: $\displaystyle \int_{\frac{1}{e}}^{e}\left|\lg x\right| dx$.

Using the integration-by-parts formula, evaluate the following definite integral: $\displaystyle \int_{0}^{1}\arccos x dx$.

Using the integration-by-parts formula, evaluate the following definite integral: $\displaystyle \int_{0}^{\sqrt{3}}x\arctan x dx$.

Using an appropriate change of variable, evaluate the following definite integral: $\displaystyle \int_{-1}^{1}\dfrac {x dx}{\sqrt{5-4x}}$.

Using an appropriate change of variable, evaluate the following definite integral: $\displaystyle \int_{0}^{a}x^{2}\sqrt{a^{2}-x^{2}} dx$ ($a>0$).

Using an appropriate change of variable, evaluate the following definite integral: $\displaystyle \int_{0}^{0.75}\dfrac {dx}{(x+1)\sqrt{x^{2}+1}}$.

Using an appropriate change of variable, evaluate the following definite integral: $\displaystyle \int_{0}^{\ln 2}\sqrt{e^{x}-1} dx$.

Using an appropriate change of variable, evaluate the following definite integral: $\displaystyle \int_{0}^{1}\dfrac {\arcsin \sqrt{x}}{\sqrt{x(1-x)}} dx$.

Suppose $x-\dfrac {1}{x}=t$; use this to compute the integral $\displaystyle \int_{-1}^{1}\dfrac {1+x^{2}}{1+x^{4}} dx$.

Given:

In the integral $\displaystyle \int_{0}^{9}x\sqrt[3]{1-x^{2}} dx$, is it permissible to set $x=\sin t$?

In the integral $\displaystyle \int_{0}^{1}\sqrt{1-x^{2}} dx$, when making the change of variable $x=\sin t$, may we take $\pi$ and $\dfrac {\pi }{2}$ as the new upper and lower limits?

Prove that if the function $f(x)$ is continuous on the closed interval $[a,b]$, then

Prove the identity:

Let $f(x)$ be a continuous function on the closed interval $[A,B]\supset [a,b]$; for $A-a\leqslant x\leqslant B-b$, find $\displaystyle \dfrac {d}{dx}\int_{a}^{b}f(x+y) dy$.

Prove that if the function $f(x)$ is continuous on the closed interval $[0,1]$, then

Prove that if the function $f(x)$ is continuous on the closed interval $[-l,l]$, then

Prove that one of the antiderivatives of an even function is an odd function, while every antiderivative of an odd function is an even function.

Introduce the new variable

Use it to compute the integral $\displaystyle \int_{\frac{1}{2}}^{2}\left(1+x-\dfrac {1}{x}\right)e^{x+\frac{1}{x}} dx$.

In the integral $\displaystyle \int_{0}^{2\pi }f(x)\cos x dx$, carry out the change of variable $\sin x=t$.

Compute the integral

where $n$ is a natural number.

Find: $\displaystyle \int_{0}^{\pi }\dfrac {x\sin x}{1+\cos^{2}x} dx$.

Let

find $\displaystyle \int_{-1}^{3}\dfrac {f’(x)}{1+f^{2}(x)} dx$.

Prove that if $f(x)$ is a continuous periodic function defined for $-\infty <x<+\infty$ with period $T$, then

where $a$ is an arbitrary number.

Prove that when $n$ is odd, the functions

are periodic functions with period $2\pi$; while when $n$ is even, each of them is the sum of a linear function and a periodic function.

Prove that the function

(where $f(x)$ is a continuous periodic function with period $T$) is, in the general case, the sum of a linear function and a periodic function.

Evaluate the following integral: $\displaystyle \int_{0}^{1}x(2-x^{2})^{12} dx$.

Evaluate the following integral: $\displaystyle \int_{-1}^{1}\dfrac {x dx}{x^{2}+x+1}$.

Evaluate the following integral: $\displaystyle \int_{1}^{e}(x\ln x)^{2} dx$.

Evaluate the following integral: $\displaystyle \int_{1}^{9}x\sqrt[3]{1-x} dx$.

Evaluate the following integral: $\displaystyle \int_{-2}^{-1}\dfrac {dx}{x\sqrt{x^{2}-1}}$.

Evaluate the following integral: $\displaystyle \int_{0}^{1}x^{15}\sqrt{1+3x^{8}} dx$.

Evaluate the following integral: $\displaystyle \int_{0}^{3}\arcsin \sqrt{\dfrac {x}{1+x}} dx$.

Evaluate the following integral: $\displaystyle \int_{0}^{2\pi }\dfrac {dx}{(2+\cos x)(3+\cos x)}$.

Evaluate the following integral: $\displaystyle \int_{0}^{2\pi }\dfrac {dx}{\sin^{4}x+\cos^{4}x}$.

Evaluate the following integral: $\displaystyle \int_{0}^{\frac{\pi }{2}}\sin x\sin 2x\sin 3x dx$.

Evaluate the following integral: $\displaystyle \int_{0}^{\pi }(x\sin x)^{2} dx$.

Evaluate the following integral: $\displaystyle \int_{0}^{\pi }e^{x}\cos^{2}x dx$.

Evaluate the following integral: $\displaystyle \int_{0}^{\ln 2}\sinh^{4}x dx$.

Using a recursion formula, compute the following integral, which depends on a parameter $n$ taking positive integer values: $\displaystyle I_{n}=\int_{0}^{\frac{\pi }{2}}\sin^{n}x dx$.

Using a recursion formula, compute the following integral, which depends on a parameter $n$ taking positive integer values: $\displaystyle I_{n}=\int_{0}^{\frac{\pi }{2}}\cos^{n}x dx$.

Using a recursion formula, compute the following integral, which depends on a parameter $n$ taking positive integer values: $\displaystyle I_{n}=\int_{0}^{\frac{\pi }{4}}\tan^{2n}x dx$.

Using a recursion formula, compute the following integral, which depends on a parameter $n$ taking positive integer values: $\displaystyle I_{n}=\int_{0}^{1}(1-x^{2})^{n} dx$.

Using a recursion formula, compute the following integral, which depends on a parameter $n$ taking positive integer values: $\displaystyle I_{n}=\int_{0}^{1}\dfrac {x^{n} dx}{\sqrt{1-x^{2}}}$.

Using a recursion formula, compute the following integral, which depends on a parameter $n$ taking positive integer values: $\displaystyle I_{n}=\int_{0}^{1}x^{m}(\ln x)^{n} dx$.

Using a recursion formula, compute the following integral, which depends on a parameter $n$ taking positive integer values: $\displaystyle I_{n}=\int_{0}^{\frac{\pi }{4}}\left(\dfrac {\sin x-\cos x}{\sin x+\cos x}\right)^{2n+1} dx$.

Using Euler's formula

prove that:

($n$ and $m$ are integers).

Prove that

($\alpha$ and $\beta$ are constants).

Using Euler's formula $\cos x=\dfrac {1}{2}(e^{ix}+e^{-ix})$, $\sin x=\dfrac {1}{2i}(e^{ix}-e^{-ix})$, compute the following integral ($m$ and $n$ are positive integers): $\displaystyle \int_{0}^{\frac{\pi }{2}}\sin^{2m}x\cos^{2n}x dx$.

Using Euler's formula $\cos x=\dfrac {1}{2}(e^{ix}+e^{-ix})$, $\sin x=\dfrac {1}{2i}(e^{ix}-e^{-ix})$, compute the following integral ($m$ and $n$ are positive integers): $\displaystyle \int_{0}^{\pi }\dfrac {\sin nx}{\sin x} dx$.

Using Euler's formula $\cos x=\dfrac {1}{2}(e^{ix}+e^{-ix})$, $\sin x=\dfrac {1}{2i}(e^{ix}-e^{-ix})$, compute the following integrals ($m$ and $n$ are positive integers): $\displaystyle \int_{0}^{\pi }\dfrac {\cos (2n+1)x}{\cos x} dx$.

Using Euler's formula $\cos x=\dfrac {1}{2}(e^{ix}+e^{-ix})$, $\sin x=\dfrac {1}{2i}(e^{ix}-e^{-ix})$, compute the following integrals ($m$ and $n$ are positive integers): $\displaystyle \int_{0}^{\pi }\cos^{n}x\cos nx dx$.