Can a human calculate this without a calculator?

[Graxum]

Homework Statement

evaluate the integral $\int 75 \sin^3(x) \cos^2 (x)dx$

Relevant Equations

u-substitution

my notebook says that we can rewrite the integral

$$\int {75\sin^3⁡(x) \cos^2⁡(x)dx}$$

as

$$\int {75 \cos^2(x)\sin(x)dx} - \int {75\sin(x)\cos^4(x)dx}$$

however, i have literally no idea how it got to this point, and i unfortunately can't really provide an "attempt at a solution" for this. If we can separate integrals with multiplication in their integrands in such a way, why don't we use this more often?

 

[Orodruin]

Hint: $\sin^2(x) + \cos^2(x) = 1$

 

[PeroK]

Homework Statement:: evaluate the integral $$\int 75 \sin^3(x) \cos^2 (x)dx$$
Relevant Equations:: u-substitution

my notebook says that we can rewrite the integral

$$\int {75\sin^3⁡(x) \cos^2⁡(x)dx}$$

as

$$\int {75 \cos^2(x)\sin(x)dx} - \int {75\sin(x)\cos^4(x)dx}$$

however, i have literally no idea how it got to this point, and i unfortunately can't really provide an "attempt at a solution" for this. If we can separate integrals with multiplication in their integrands in such a way, why don't we use this more often?

This human can't even read it without proper Latex rendering!

 

[Orodruin]

This human can't even read it without proper Latex rendering!

TIL: My humanity was revoked.

 

[Graxum]

This human can't even read it without proper Latex rendering!

ey, the preview seems to be messed up. whenever i click "preview" it doesn't change anything and i have to reload the page in order to get it to work, and when i turn off preview mode i lose a bunch of stuff i wrote.

 

[PeroK]

ey, the preview seems to be messed up. whenever i click "preview" it doesn't change anything and i have to reload the page in order to get it to work, and when i turn off preview mode i lose a bunch of stuff i wrote.

I know, it's a problem with the software. But, then, the computer is never wrong apparently!

 

[berkeman]

ey, the preview seems to be messed up. whenever i click "preview" it doesn't change anything and i have to reload the page in order to get it to work, and when i turn off preview mode i lose a bunch of stuff i wrote.

I fixed your LaTeX in the OP for you. You were using single-$ delimiters instead of double-$.

 

[PeroK]

I fixed your LaTeX in the OP for you. You were using single-$ delimiters instead of double-$.

It's amazing what you can get for a few dollars more!

 

[vela]

If we can separate integrals with multiplication in their integrands in such a way, why don't we use this more often?

This question strikes me as odd considering you said you had "literally no idea how it got to this point." What exactly is "this"? You'll see using @Orodruin's hint that they used the usual properties of integration. (I'm not sure why they would bother splitting the original integral into two integrals. It seems to be more work.)

Have you checked your textbook for doing these types of integrals? There should be a section on how to attack integrals of this form.

 

[Steelgrip]

Homework Statement:: evaluate the integral $\int 75 \sin^3(x) \cos^2 (x)dx$
Relevant Equations:: u-substitution

my notebook says that we can rewrite the integral

$$\int {75\sin^3⁡(x) \cos^2⁡(x)dx}$$

as

$$\int {75 \cos^2(x)\sin(x)dx} - \int {75\sin(x)\cos^4(x)dx}$$

however, i have literally no idea how it got to this point, and i unfortunately can't really provide an "attempt at a solution" for this. If we can separate integrals with multiplication in their integrands in such a way, why don't we use this more often?

Heyy that seems too easy of a problem, that 75 is just a multiple right?
I can't wait to provide an image solution just clear my query

 

[berkeman]

Heyy that seems too easy of a problem, that 75 is just a multiple right?
I can't wait to provide an image solution just clear my query

Please don't post solutions to schoolwork questions, unless you were the person who started the thread and have figured it out. Also, please don't post "images" of math -- see the "LaTeX Guide" link below the Edit window to learn how to post math equations here at PF. Thank you.

 

[Steelgrip]

Please don't post solutions to schoolwork questions, unless you were the person who started the thread and have figured it out. Also, please don't post "images" of math -- see the "LaTeX Guide" link below the Edit window to learn how to post math equations here at PF. Thank you.

Okay, I sure. I didn't know that.