2nalfq8

2022-07-06

Antiderivatives and definite integrals

Compute the following integral:

$I={\int}_{0}^{1}{\textstyle (}{\int}_{1}^{x}\frac{1}{1+{t}^{2}}dt{\textstyle )}dx$

Motivation: From time to time, i intend to post here challenging questions -together with proposed answers/solutions- which i frequently give for practise to some of my students (in an introductory calculus course).

I am doing this with the understanding that this is not only accepted but furthermore encouraged by the community. The benefits can be twofold: on the one hand this contributes to building a library with well-posed questions and reliable answers (to be used in-class or online) and on the other hand i always hope that this may result in improving (or sometimes correcting) the proposed questions/answers and discovering new approaches/solutions from other users as well.

Compute the following integral:

$I={\int}_{0}^{1}{\textstyle (}{\int}_{1}^{x}\frac{1}{1+{t}^{2}}dt{\textstyle )}dx$

Motivation: From time to time, i intend to post here challenging questions -together with proposed answers/solutions- which i frequently give for practise to some of my students (in an introductory calculus course).

I am doing this with the understanding that this is not only accepted but furthermore encouraged by the community. The benefits can be twofold: on the one hand this contributes to building a library with well-posed questions and reliable answers (to be used in-class or online) and on the other hand i always hope that this may result in improving (or sometimes correcting) the proposed questions/answers and discovering new approaches/solutions from other users as well.

Maggie Bowman

Beginner2022-07-07Added 14 answers

Explanation:

$\begin{array}{rl}{\displaystyle {\int}_{0}^{1}{\int}_{1}^{x}\frac{dt}{1+{t}^{2}}\cdot dx}& ={\int}_{0}^{1}(\mathrm{arctan}(x)-\frac{\pi}{4})dx\\ & ={\textstyle [}x\mathrm{arctan}(x)-\frac{1}{2}\mathrm{ln}(1+{x}^{2})-\frac{\pi}{4}x{{\textstyle ]}}_{0}^{1}=-\frac{1}{2}\mathrm{ln}(2)\end{array}$

$\begin{array}{rl}{\displaystyle {\int}_{0}^{1}{\int}_{1}^{x}\frac{dt}{1+{t}^{2}}\cdot dx}& ={\int}_{0}^{1}(\mathrm{arctan}(x)-\frac{\pi}{4})dx\\ & ={\textstyle [}x\mathrm{arctan}(x)-\frac{1}{2}\mathrm{ln}(1+{x}^{2})-\frac{\pi}{4}x{{\textstyle ]}}_{0}^{1}=-\frac{1}{2}\mathrm{ln}(2)\end{array}$

dream13rxs

Beginner2022-07-08Added 4 answers

Step 1

Let $f(t)=\frac{1}{1+{t}^{2}}$, $t\in \mathbb{R}$ and $F(x)={\int}_{1}^{x}\frac{1}{1+{t}^{2}}dt={\int}_{1}^{x}f(t)dt$, $t\in \mathbb{R}$

Step 2

Consequently, ${F}^{\prime}(x)=f(x)$, $x\in \mathbb{R}$ and:

$I={\int}_{0}^{1}{\textstyle (}{\int}_{1}^{x}\frac{1}{1+{t}^{2}}dt{\textstyle )}dx={\int}_{0}^{1}F(x)dx={\int}_{0}^{1}(x{)}^{\prime}F(x)dx=$

$=[xF(x){]}_{0}^{1}-{\int}_{0}^{1}x{F}^{\prime}(x)dx=F(1)-{\int}_{0}^{1}xf(x)dx=0-{\int}_{0}^{1}\frac{x}{1+{x}^{2}}dx=$

$=-\frac{1}{2}{\int}_{0}^{1}\frac{(1+{x}^{2}{)}^{\prime}}{1+{x}^{2}}dx=-\frac{1}{2}[\mathrm{ln}(1+{x}^{2}){]}_{0}^{1}=-\frac{\mathrm{ln}2}{2}$

Let $f(t)=\frac{1}{1+{t}^{2}}$, $t\in \mathbb{R}$ and $F(x)={\int}_{1}^{x}\frac{1}{1+{t}^{2}}dt={\int}_{1}^{x}f(t)dt$, $t\in \mathbb{R}$

Step 2

Consequently, ${F}^{\prime}(x)=f(x)$, $x\in \mathbb{R}$ and:

$I={\int}_{0}^{1}{\textstyle (}{\int}_{1}^{x}\frac{1}{1+{t}^{2}}dt{\textstyle )}dx={\int}_{0}^{1}F(x)dx={\int}_{0}^{1}(x{)}^{\prime}F(x)dx=$

$=[xF(x){]}_{0}^{1}-{\int}_{0}^{1}x{F}^{\prime}(x)dx=F(1)-{\int}_{0}^{1}xf(x)dx=0-{\int}_{0}^{1}\frac{x}{1+{x}^{2}}dx=$

$=-\frac{1}{2}{\int}_{0}^{1}\frac{(1+{x}^{2}{)}^{\prime}}{1+{x}^{2}}dx=-\frac{1}{2}[\mathrm{ln}(1+{x}^{2}){]}_{0}^{1}=-\frac{\mathrm{ln}2}{2}$

Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function

$f(x,y)={x}^{3}-6xy+8{y}^{3}$ $\frac{1}{\mathrm{sec}(x)}$ in derivative?

What is the derivative of $\mathrm{ln}(x+1)$?

What is the limit of $e}^{-x$ as $x\to \infty$?

What is the derivative of $f\left(x\right)={5}^{\mathrm{ln}x}$?

What is the derivative of $e}^{-2x$?

How to find $lim\frac{{e}^{t}-1}{t}$ as $t\to 0$ using l'Hospital's Rule?

What is the integral of $\sqrt{9-{x}^{2}}$?

What is the derivative of $f\left(x\right)=\mathrm{ln}\left[{x}^{9}{(x+3)}^{6}{({x}^{2}+7)}^{5}\right]$ ?

What Is the common difference or common ratio of the sequence 2, 5, 8, 11...?

How to find the derivative of $y={e}^{5x}$?

How to evaluate the limit $\frac{\mathrm{sin}\left(5x\right)}{x}$ as x approaches 0?

How to find derivatives of parametric functions?

What is the antiderivative of $-5{e}^{x-1}$?

How to evaluate: indefinite integral $\frac{1+x}{1+{x}^{2}}dx$?