Integration by Parts: Solving ∫sin²x dx with Ease

[annamariesmit]

how would one integrate by parts the following:
$$\int sin^2xdx$$

thanks!

 

[rsm]

answer

hi
use the fact that sin^2 x = (1-cos2x)/2
from the formula cos2x=1-2sin^2 x

Tell me how you wrote that equation

 

[malawi_glenn]

[ tex]\int sin^2xdx[/tex ]

Click on the image (equation) and the TeX code comes up. See also:
https://www.physicsforums.com/showthread.php?t=8997

Aslo this is not a forum for homeworks, https://www.physicsforums.com/showthread.php?t=44101

Post questions regarding homework and similar in the homework section.

 

[HallsofIvy]

how would one integrate by parts the following:
$$\int sin^2xdx$$

thanks!

Are you required to use integration by parts? As rsm said, there are simple and standard substitutions for $sin^2(x)$ and $cos^2(x)$.

If you are required to use integration by parts, then, since integration by parts requires a product, the obvious thing to do it write this as a product:
$$\int sin^2(x) dx= \int (sin(x))(sin(x) dx)$$
Let u= sin(x) and let dv= sin(x) dx. Then du= cos(x)dx and v= -cos(x)
$$\int sin^2 x dx= -sin(x)cos(x)+ \int cos^2(x) dx$$
Now do the same thing with that integral. Of course, what happens is you will get back to your original $\int sin^2(x) dx$- but with a lot of other things. Solve that equation algebraically for $\int sin^2(x)dx$