differnetiate
-sinx*cosx dx
would i use substitution of
u=cosx
du=-sinx
so it would be u du and therefore ux+c
so it would be cosx+c? RIGHT?
am i on the right track?
Yes, you are on the right track. To differentiate the given expression, you can indeed use the substitution method. Let's go through the steps:
1. Let u = cos(x).
Here, u is the new variable we will use for substitution.
2. Taking the derivative of both sides with respect to x, we get:
du/dx = -sin(x).
Notice that du/dx is equivalent to (d/dx)(u).
3. Rearrange the equation from step 2 to get dx in terms of du:
dx = du/(-sin(x)).
Notice that, since u = cos(x), sin(x) = sqrt(1 - u^2).
4. Replace dx in the original expression with du/(-sin(x)):
-sin(x) * cos(x) dx = -u(du/sqrt(1 - u^2)).
5. Simplify the expression:
-u(du/sqrt(1 - u^2)) = -u * du / sqrt(1 - u^2).
6. Integrate the simplified expression with respect to u:
∫ (-u * du / sqrt(1 - u^2)) = -√(1 - u^2) + C.
Here, C represents the constant of integration.
7. Substitute u back to its original form, cos(x):
-√(1 - cos^2(x)) + C.
So, the final answer is -√(1 - cos^2(x)) + C. You were close, but the correct answer involves a square root term due to the substitution.