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Q&A Am I taking the antiderivative of |x| correctly?

1 answer · posted 7mo ago by CrSb0001‭ · edited 4mo ago by The Amplitwist‭

Question calculus integration recreational-mathematics

#4: Post edited by $user avatar$ The Amplitwist‭ · 2024-07-22T15:47:59Z (4 months ago)
removed extraneous []() introduced in the previous edit

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Am I taking the antiderivative of |x| correctly?

Edit: forgot +C, sorry
So I remember seeing [this][1] video by Blackpenredpen on the Youtube homepage around 3 months ago where he was showing failed attempts of him trying to figure out how to get$$\int\sqrt{\sin^2(x)} \,dx=\int\left|\sin(x)\right| \, dx=-\cot(x)\left|\sin(x)\right|+c$$without piecewise integration.
While I do have my own attempts at this that I've done over the course of 2 months (by attempting IBP or using $u$-substitution), I think these attempts have failed because I don't really understand how to even evaluate$$\int|x|\,dx$$So I think that if I can integrate this, I can integrate$$\int\left|\sin(x)\right| \, dx$$ using a similar method. Here is my attempt at doing so:
We have that$$\int|x|\,dx=\int1\cdot|x| \,dx$$Now, we can rewrite this as$$\int\left(\frac d{dx}x\right)|x|\,dx$$and now we can use IBP.
Here's a brief overview of IBP for anyone who doesn't remember it:
>IBP, which stands for `Integration By Parts`, is a method of integration that states that for functions `f(x)` and `g(x)`,$$\int f(x)\,dg(x)=f(x)g(x)-\int g(x)\,df(x)$$(here $dg(x)$ and $df(x)$ means $\frac d{dx}g(x)$ and $\frac d{dx}f(x)$ respectively. I understand the notation is disliked a lot) So what I think it is trying to show here in this context is that if we can rewrite our integral as a product of a 2 functions, and we know the antiderivative of one of them, we can use that to find the antiderivative of the other function.
So now we can use IBP on our integral$$I=\int\left(\frac d{dx}x ight) |x| \, dx$$using$$f(x)=|x|$$$$g(x)=x$$to get$$I=\int\left(\frac d{dx}x ight)|x|=x|x|-\int x \cdot \left(\frac{|x|}x ight)\,dx$$$$=x|x|-\int|x|\,dx$$$$I=x|x|-\int|x| \, []()dx\implies I=x|x|-I$$$$\implies 2I=x|x|\implies I=\int|x|\,dx=\frac{x|x|}2$$So we have that the antiderivative of $|x|$ is $\frac{x|x|}2+C$.
However, my question is: Am I taking the antiderivative of $|x|$ correctly, or how would I take it?
[1]:https://youtu.be/-kJheoMb2ys?si=ch6kGADWbrKxcMOx

Edit: forgot +C, sorry
So I remember seeing [this][1] video by Blackpenredpen on the Youtube homepage around 3 months ago where he was showing failed attempts of him trying to figure out how to get$$\int\sqrt{\sin^2(x)} \,dx=\int\left|\sin(x)\right| \, dx=-\cot(x)\left|\sin(x)\right|+c$$without piecewise integration.
While I do have my own attempts at this that I've done over the course of 2 months (by attempting IBP or using $u$-substitution), I think these attempts have failed because I don't really understand how to even evaluate$$\int|x|\,dx$$So I think that if I can integrate this, I can integrate$$\int\left|\sin(x)\right| \, dx$$ using a similar method. Here is my attempt at doing so:
We have that$$\int|x|\,dx=\int1\cdot|x| \,dx$$Now, we can rewrite this as$$\int\left(\frac d{dx}x\right)|x|\,dx$$and now we can use IBP.
Here's a brief overview of IBP for anyone who doesn't remember it:
>IBP, which stands for `Integration By Parts`, is a method of integration that states that for functions `f(x)` and `g(x)`,$$\int f(x)\,dg(x)=f(x)g(x)-\int g(x)\,df(x)$$(here $dg(x)$ and $df(x)$ means $\frac d{dx}g(x)$ and $\frac d{dx}f(x)$ respectively. I understand the notation is disliked a lot) So what I think it is trying to show here in this context is that if we can rewrite our integral as a product of a 2 functions, and we know the antiderivative of one of them, we can use that to find the antiderivative of the other function.
So now we can use IBP on our integral$$I=\int\left(\frac d{dx}x ight) |x| \, dx$$using$$f(x)=|x|$$$$g(x)=x$$to get$$I=\int\left(\frac d{dx}x ight)|x|=x|x|-\int x \cdot \left(\frac{|x|}x ight)\,dx$$$$=x|x|-\int|x|\,dx$$$$I=x|x|-\int|x| \, dx\implies I=x|x|-I$$$$\implies 2I=x|x|\implies I=\int|x|\,dx=\frac{x|x|}2$$So we have that the antiderivative of $|x|$ is $\frac{x|x|}2+C$.
However, my question is: Am I taking the antiderivative of $|x|$ correctly, or how would I take it?
[1]:https://youtu.be/-kJheoMb2ys?si=ch6kGADWbrKxcMOx

#3: Post edited by $user avatar$ Michael Hardy‭ · 2024-07-18T09:12:32Z (4 months ago)
Proper horizontal spacing between f(x) and dx. Also, without \left and \right, the absolute value signs lack the right horizontal spacing when \sin is used.

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Am I taking the antiderivative of |x| correctly?

Edit: forgot +C, sorry
So I remember seeing [this][1] video by Blackpenredpen on the Youtube homepage around 3 months ago where he was showing failed attempts of him trying to figure out how to get$$\int\sqrt{\sin^2(x)}dx=\int|\sin(x)|dx=-\cot(x)|\sin(x)|+c$$without piecewise integration.
While I do have my own attempts at this that I've done over the course of 2 months (by attempting IBP or using $u$-substitution), I think these attempts have failed because I don't really understand how to even evaluate$$\int|x|dx$$So I think that if I can integrate this, I can integrate$$\int|\sin(x)|dx$$using a similar method. Here is my attempt at doing so:
~~We have that$$\int|x|dx=\int1\cdot|x|dx$$Now, we can rewrite this as$$\int\left(\frac d{dx}x ight)|x|dx$$and now we can use IBP.~~
Here's a brief overview of IBP for anyone who doesn't remember it:
>IBP, which stands for `Integration By Parts`, is a method of integration that states that for functions `f(x)` and `g(x)`,$$\int f(x)dg(x)=f(x)g(x)-\int g(x)df(x)$$(here $dg(x)$ and $df(x)$ means $\frac d{dx}g(x)$ and $\frac d{dx}f(x)$ respectively. I understand the notation is disliked a lot) So what I think it is trying to show here in this context is that if we can rewrite our integral as a product of a 2 functions, and we know the antiderivative of one of them, we can use that to find the antiderivative of the other function.
So now we can use IBP on our integral$$I=\int\left(\frac d{dx}x ight)|x|dx$$using$$f(x)=|x|$$$$g(x)=x$$to get$$I=\int\left(\frac d{dx}x ight)|x|=x|x|-\int x\cdot\left(\frac{|x|}x ight)dx$$$$=x|x|-\int|x|dx$$$$I=x|x|-\int|x|dx\implies I=x|x|-I$$$$\implies 2I=x|x|\implies I=\int|x|dx=\frac{x|x|}2$$So we have that the antiderivative of $|x|$ is $\frac{x|x|}2+C$.
~~However, my question is: Am I taking the antiderivative of |x| correctly, or how would I take it?~~
[1]:https://youtu.be/-kJheoMb2ys?si=ch6kGADWbrKxcMOx

Edit: forgot +C, sorry
So I remember seeing [this][1] video by Blackpenredpen on the Youtube homepage around 3 months ago where he was showing failed attempts of him trying to figure out how to get$$\int\sqrt{\sin^2(x)} \,dx=\int\left|\sin(x)\right| \, dx=-\cot(x)\left|\sin(x)\right|+c$$without piecewise integration.
While I do have my own attempts at this that I've done over the course of 2 months (by attempting IBP or using $u$-substitution), I think these attempts have failed because I don't really understand how to even evaluate$$\int|x|\,dx$$So I think that if I can integrate this, I can integrate$$\int\left|\sin(x)\right| \, dx$$ using a similar method. Here is my attempt at doing so:
We have that$$\int|x|\,dx=\int1\cdot|x| \,dx$$Now, we can rewrite this as$$\int\left(\frac d{dx}x ight)|x|\,dx$$and now we can use IBP.
Here's a brief overview of IBP for anyone who doesn't remember it:
>IBP, which stands for `Integration By Parts`, is a method of integration that states that for functions `f(x)` and `g(x)`,$$\int f(x)\,dg(x)=f(x)g(x)-\int g(x)\,df(x)$$(here $dg(x)$ and $df(x)$ means $\frac d{dx}g(x)$ and $\frac d{dx}f(x)$ respectively. I understand the notation is disliked a lot) So what I think it is trying to show here in this context is that if we can rewrite our integral as a product of a 2 functions, and we know the antiderivative of one of them, we can use that to find the antiderivative of the other function.
So now we can use IBP on our integral$$I=\int\left(\frac d{dx}x ight) |x| \, dx$$using$$f(x)=|x|$$$$g(x)=x$$to get$$I=\int\left(\frac d{dx}x ight)|x|=x|x|-\int x \cdot \left(\frac{|x|}x ight)\,dx$$$$=x|x|-\int|x|\,dx$$$$I=x|x|-\int|x| \, []()dx\implies I=x|x|-I$$$$\implies 2I=x|x|\implies I=\int|x|\,dx=\frac{x|x|}2$$So we have that the antiderivative of $|x|$ is $\frac{x|x|}2+C$.
However, my question is: Am I taking the antiderivative of $|x|$ correctly, or how would I take it?
[1]:https://youtu.be/-kJheoMb2ys?si=ch6kGADWbrKxcMOx

#2: Post edited by $user avatar$ CrSb0001‭ · 2024-05-09T16:39:09Z (7 months ago)
forgot +C

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So I remember seeing [this][1] video by Blackpenredpen on the Youtube homepage around 3 months ago where he was showing failed attempts of him trying to figure out how to get$$\int\sqrt{\sin^2(x)}dx=\int|\sin(x)|dx=-\cot(x)|\sin(x)|+c$$without piecewise integration.
While I do have my own attempts at this that I've done over the course of 2 months (by attempting IBP or using $u$-substitution), I think these attempts have failed because I don't really understand how to even evaluate$$\int|x|dx$$So I think that if I can integrate this, I can integrate$$\int|\sin(x)|dx$$using a similar method. Here is my attempt at doing so:
We have that$$\int|x|dx=\int1\cdot|x|dx$$Now, we can rewrite this as$$\int\left(\frac d{dx}x\right)|x|dx$$and now we can use IBP.
Here's a brief overview of IBP for anyone who doesn't remember it:
>IBP, which stands for `Integration By Parts`, is a method of integration that states that for functions `f(x)` and `g(x)`,$$\int f(x)dg(x)=f(x)g(x)-\int g(x)df(x)$$(here $dg(x)$ and $df(x)$ means $\frac d{dx}g(x)$ and $\frac d{dx}f(x)$ respectively. I understand the notation is disliked a lot) So what I think it is trying to show here in this context is that if we can rewrite our integral as a product of a 2 functions, and we know the antiderivative of one of them, we can use that to find the antiderivative of the other function.
So now we can use IBP on our integral$$I=\int\left(\frac d{dx}x ight)|x|dx$$using$$f(x)=|x|$$$$g(x)=x$$to get$$I=\int\left(\frac d{dx}x ight)|x|=x|x|-\int x\cdot\left(\frac{|x|}x ight)dx$$$$=x|x|-\int|x|dx$$$$I=x|x|-\int|x|dx\implies I=x|x|-I$$$$\implies 2I=x|x|\implies I=\int|x|dx=\frac{x|x|}2$$So we have that the antiderivative of $|x|$ is $\frac{x|x|}2$.
However, my question is: Am I taking the antiderivative of |x| correctly, or how would I take it?
[1]:https://youtu.be/-kJheoMb2ys?si=ch6kGADWbrKxcMOx

Edit: forgot +C, sorry
So I remember seeing [this][1] video by Blackpenredpen on the Youtube homepage around 3 months ago where he was showing failed attempts of him trying to figure out how to get$$\int\sqrt{\sin^2(x)}dx=\int|\sin(x)|dx=-\cot(x)|\sin(x)|+c$$without piecewise integration.
While I do have my own attempts at this that I've done over the course of 2 months (by attempting IBP or using $u$-substitution), I think these attempts have failed because I don't really understand how to even evaluate$$\int|x|dx$$So I think that if I can integrate this, I can integrate$$\int|\sin(x)|dx$$using a similar method. Here is my attempt at doing so:
We have that$$\int|x|dx=\int1\cdot|x|dx$$Now, we can rewrite this as$$\int\left(\frac d{dx}x\right)|x|dx$$and now we can use IBP.
Here's a brief overview of IBP for anyone who doesn't remember it:
>IBP, which stands for `Integration By Parts`, is a method of integration that states that for functions `f(x)` and `g(x)`,$$\int f(x)dg(x)=f(x)g(x)-\int g(x)df(x)$$(here $dg(x)$ and $df(x)$ means $\frac d{dx}g(x)$ and $\frac d{dx}f(x)$ respectively. I understand the notation is disliked a lot) So what I think it is trying to show here in this context is that if we can rewrite our integral as a product of a 2 functions, and we know the antiderivative of one of them, we can use that to find the antiderivative of the other function.
So now we can use IBP on our integral$$I=\int\left(\frac d{dx}x ight)|x|dx$$using$$f(x)=|x|$$$$g(x)=x$$to get$$I=\int\left(\frac d{dx}x ight)|x|=x|x|-\int x\cdot\left(\frac{|x|}x ight)dx$$$$=x|x|-\int|x|dx$$$$I=x|x|-\int|x|dx\implies I=x|x|-I$$$$\implies 2I=x|x|\implies I=\int|x|dx=\frac{x|x|}2$$So we have that the antiderivative of $|x|$ is $\frac{x|x|}2+C$.
However, my question is: Am I taking the antiderivative of |x| correctly, or how would I take it?
[1]:https://youtu.be/-kJheoMb2ys?si=ch6kGADWbrKxcMOx

#1: Initial revision by $user avatar$ CrSb0001‭ · 2024-05-09T14:19:11Z (7 months ago)

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Am I taking the antiderivative of |x| correctly?

So I remember seeing [this][1] video by Blackpenredpen on the Youtube homepage around 3 months ago where he was showing failed attempts of him trying to figure out how to get$$\int\sqrt{\sin^2(x)}dx=\int|\sin(x)|dx=-\cot(x)|\sin(x)|+c$$without piecewise integration.

While I do have my own attempts at this that I've done over the course of 2 months (by attempting IBP or using $u$-substitution), I think these attempts have failed because I don't really understand how to even evaluate$$\int|x|dx$$So I think that if I can integrate this, I can integrate$$\int|\sin(x)|dx$$using a similar method. Here is my attempt at doing so:

We have that$$\int|x|dx=\int1\cdot|x|dx$$Now, we can rewrite this as$$\int\left(\frac d{dx}x\right)|x|dx$$and now we can use IBP.

Here's a brief overview of IBP for anyone who doesn't remember it:
>IBP, which stands for `Integration By Parts`, is a method of integration that states that for functions `f(x)` and `g(x)`,$$\int f(x)dg(x)=f(x)g(x)-\int g(x)df(x)$$<sup>(here $dg(x)$ and $df(x)$ means $\frac d{dx}g(x)$ and $\frac d{dx}f(x)$ respectively. I understand the notation is disliked a lot)</sup><br>So what I think it is trying to show here in this context is that if we can rewrite our integral as a product of a 2 functions, and we know the antiderivative of one of them, we can use that to find the antiderivative of the other function.

So now we can use IBP on our integral$$I=\int\left(\frac d{dx}x\right)|x|dx$$using$$f(x)=|x|$$$$g(x)=x$$to get$$I=\int\left(\frac d{dx}x\right)|x|=x|x|-\int x\cdot\left(\frac{|x|}x\right)dx$$$$=x|x|-\int|x|dx$$$$I=x|x|-\int|x|dx\implies I=x|x|-I$$$$\implies 2I=x|x|\implies I=\int|x|dx=\frac{x|x|}2$$So we have that the antiderivative of $|x|$ is $\frac{x|x|}2$.

However, my question is: Am I taking the antiderivative of |x| correctly, or how would I take it?

[1]:https://youtu.be/-kJheoMb2ys?si=ch6kGADWbrKxcMOx

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