Methods for Computing Definite Integrals Using Indefinite Integrals

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:

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

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): $\displaystyle \int_{0}^{\pi }\cos^{n}x\cos nx dx$.

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): $\displaystyle \int_{0}^{\pi }\sin^{n}x\sin nx dx$.

Find the following integrals ($n$ is a natural number): $\displaystyle \int_{0}^{\pi }\sin^{n-1}x\cos (n+1)x dx$.

Find the following integrals ($n$ is a natural number): $\displaystyle \int_{0}^{\pi }\cos^{n-1}x\sin (n+1)x dx$.

Find the following integrals ($n$ is a natural number): $\displaystyle \int_{0}^{2\pi }e^{-ax}\cos^{2n}x dx$.

Find the following integrals ($n$ is a natural number): $\displaystyle \int_{0}^{\frac{\pi }{2}}\ln \cos x\cdot \cos 2nx dx$.

Using integration by parts repeatedly, compute Euler's integral:

where $m$ and $n$ are positive integers.

The Legendre polynomial $P_{n}(x)$ is defined by the following formula:

Prove that

Suppose the function $f(x)$ is integrable on $[a,b]$, and the function $F(x)$ satisfies $F'(x)=f(x)$ everywhere on $[a,b]$ except at finitely many points $c_{i}$ ($i=1,\ldots ,p$) and the points $a$ and $b$, while at these excluded finitely many points $F(x)$ has a discontinuity of the first kind (a generalized antiderivative). Prove that

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

is an antiderivative of $f(x)$. Prove that the function $F(x)$ is continuous and that the equality

holds at every point where $f(x)$ is continuous; what is the derivative of $F(x)$ at points where $f(x)$ is discontinuous?

Find the antiderivatives of the following bounded discontinuous functions: $\displaystyle \int \operatorname {sgn}x dx$.

Find the antiderivatives of the following bounded discontinuous functions: $\displaystyle \int \operatorname {sgn}(\sin x) dx$.

Find the antiderivatives of the following bounded discontinuous functions: $\displaystyle \int \left[x\right] dx$ ($x\geqslant 0$).

Find the antiderivatives of the following bounded discontinuous functions: $\displaystyle \int x\left[x\right] dx$ ($x\geqslant 0$).

Find the antiderivatives of the following bounded discontinuous functions: $\displaystyle \int (-1)^{\left[x\right]} dx$.

Find the antiderivatives of the following bounded discontinuous functions: $\displaystyle \int_{0}^{x}f(x) dx$, where $f(x)=\begin{cases} 1, & \text{if }\left|x\right|<l , \\ 0, & \text{if }\left|x\right|\geqslant l . \end{cases}$

Compute the following definite integrals of bounded discontinuous functions: $\displaystyle \int_{0}^{3}\operatorname {sgn}(x-x^{3}) dx$.

Compute the following definite integrals of bounded discontinuous functions: $\displaystyle \int_{0}^{2}\left[e^{x}\right] dx$.

Compute the following definite integrals of bounded discontinuous functions: $\displaystyle \int_{0}^{6}\left[x\right]\sin \dfrac {\pi x}{6} dx$.

Compute the following definite integrals of bounded discontinuous functions: $\displaystyle \int_{0}^{\pi }x\operatorname {sgn}(\cos x) dx$.

Compute the following definite integrals of bounded discontinuous functions: $\displaystyle \int_{1}^{n+1}\left[\ln (x)\right] dx$, where $n$ is a natural number.

Compute the following definite integrals of bounded discontinuous functions: $\displaystyle \int_{0}^{1}\operatorname {sgn}(\sin (\ln x)) dx$.

Find $\displaystyle \int_{E}\left|\cos x\right|\sqrt{\sin x} dx$, where $E$ is the set of all values in the closed interval $[0,4\pi ]$ for which the integrand is meaningful.