TomLikesPhysics 3 years ago If I want to integrate (sinx)^2 using the method of substitution than what are my function g and g`? x would be pointless because than I would just replace the x with a t but sinx also does not seem to work because there is no cosx.

1. NotSObright

Replace cos x by sqrt(1-t^2)

2. NotSObright

Put sin x = t

3. NotSObright

But going by this method is possible and feasible but not smart

4. NotSObright

$\int\limits_{}^{}\frac{t^2}{\sqrt{1-t^2}}dt$

5. NotSObright

$-\int\limits_{}^{}\sqrt{1-t^2}-\frac{1}{\sqrt{1-t^2}}$

6. TomLikesPhysics

How do you get from one term (the t^2 over sth.) to the two separate terms? At first I followed my instincts and got to the t^2 over such and such term but I thought this is a dead-end.

7. Mani_Jha

$\int\limits_{}^{}t ^{2}/\sqrt{1-t ^{2}}=\int\limits_{}^{}-(1-t ^{2}-1)/\sqrt{1-t ^{2}}$ =$-\int\limits_{}^{}(1-t ^{2})/\sqrt{1-t ^{2}}-1/\sqrt{1-t ^{2}}$. That's how he got there

8. TomLikesPhysics

Ok, that would have never come to my mind. Thank you for your explanation. I just figured out that it is quite easy to do this integral if I write instead of (sinx)^2 just 0.5-0.5*cos2x.