- #1

mcastillo356

Gold Member

- 560

- 267

- TL;DR Summary
- I'm concerned with the integral itself, but also with the possible notations for the domain.

Hi PF,

$$\int \cos ax\,dx,\quad a\in{\mathbb R-\{0\}}\quad x\in{\mathbb{R}}$$

Let's make

$$u=ax,\quad du=adx$$

and apply $$\int \cos u\,du=\sin u+C$$

$$\frac{1}{a}\int \cos ax\,adx=\frac{1}{a}\sin u+C$$

Substituting the definition of u

$$=\frac{1}{a}\sin ax+C$$

Doubts:

(i) Have I written well the integration steps? It is based on a tutorial from YouTube.

(ii) Domain of the integral is right?

(iii) ##\mathbb R-\{0\}\Leftrightarrow{\mathbb R\setminus 0}##?

(iv) Anything missing or to suggest?

Attempt

(i) It's right. A copy and paste from a video to this post.

(ii) Zero must be excluded from the domain; a becomes the denominator of a fraction when evaluating

(iii) I'm sure of the righthanded equivalence; and think I've seen lefthand notation, but I quick search on the textbook I think I read it is not been successful.

Best wishes!

PD: Post without preview

$$\int \cos ax\,dx,\quad a\in{\mathbb R-\{0\}}\quad x\in{\mathbb{R}}$$

Let's make

$$u=ax,\quad du=adx$$

and apply $$\int \cos u\,du=\sin u+C$$

$$\frac{1}{a}\int \cos ax\,adx=\frac{1}{a}\sin u+C$$

Substituting the definition of u

$$=\frac{1}{a}\sin ax+C$$

Doubts:

(i) Have I written well the integration steps? It is based on a tutorial from YouTube.

(ii) Domain of the integral is right?

(iii) ##\mathbb R-\{0\}\Leftrightarrow{\mathbb R\setminus 0}##?

(iv) Anything missing or to suggest?

Attempt

(i) It's right. A copy and paste from a video to this post.

(ii) Zero must be excluded from the domain; a becomes the denominator of a fraction when evaluating

(iii) I'm sure of the righthanded equivalence; and think I've seen lefthand notation, but I quick search on the textbook I think I read it is not been successful.

Best wishes!

PD: Post without preview

Last edited: