Can a human calculate this without a calculator?

In summary: Thanks for the heads up!In summary, my notebook says that we can rewrite the integral$$\int {75\sin^3⁡(x) \cos^2⁡(x)dx}$$as$$\int {75 \cos^2(x)\sin(x)dx} - \int {75\sin(x)\cos^4(x)dx}$$
  • #1
Graxum
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0
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 seperate integrals with multiplication in their integrands in such a way, why don't we use this more often?
 
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  • #2
Hint: ##\sin^2(x) + \cos^2(x) = 1##
 
  • #3
Graxum said:
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 seperate 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!
 
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  • #4
PeroK said:
This human can't even read it without proper Latex rendering!
TIL: My humanity was revoked. 🤔
 
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Likes berkeman
  • #5
PeroK said:
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.
 
  • #6
Graxum said:
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!
 
  • #7
Graxum said:
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-$. :wink:
 
  • #8
berkeman said:
I fixed your LaTeX in the OP for you. You were using single-$ delimiters instead of double-$. :wink:
It's amazing what you can get for a few dollars more!
 
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  • #9
Graxum said:
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.
 
  • #10
Graxum said:
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 seperate 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
 
  • #11
Steelgrip said:
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.
 
  • #12
berkeman said:
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.
 
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FAQ: Can a human calculate this without a calculator?

1. Can a human really do complex calculations without a calculator?

Yes, humans are capable of doing complex calculations without a calculator. Our brains are powerful tools that can process and manipulate numbers using various mathematical strategies.

2. What kind of calculations can humans do without a calculator?

Humans can do basic arithmetic operations such as addition, subtraction, multiplication, and division without a calculator. We can also perform more complex calculations involving fractions, decimals, and percentages.

3. How accurate are human calculations compared to those done with a calculator?

The accuracy of human calculations depends on the individual's mathematical skills and the complexity of the calculation. In general, calculators are more precise and less prone to errors compared to human calculations.

4. Is it faster to use a calculator or do calculations by hand?

For simple calculations, it may be faster to use a calculator. However, for more complex calculations, it may take longer to input the numbers and functions into a calculator compared to doing it by hand. Additionally, our brains can process information faster than we can press buttons on a calculator.

5. Are there any benefits to doing calculations without a calculator?

Yes, there are several benefits to doing calculations without a calculator. It can improve our mental math skills, increase our understanding of mathematical concepts, and help us become more efficient problem solvers. It also allows us to check the accuracy of our calculations and develop critical thinking skills.

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