Differentiate sin(cubed)x with respect to x.
Hence �find (integration sign) sin(squared) x cosx dx.
ALSO
Find the equation of the tangent to the curve y = 2 sin(x-(pie divided by 6)) at the point where x = (pie divided by 3) .
To differentiate sin^3(x) with respect to x, we can use the chain rule. Let's break it down step by step:
First, recall that the chain rule states that if we have a function of the form f(g(x)), then the derivative with respect to x is given by f'(g(x)) * g'(x).
In this case, our function is sin^3(x), and the inner function is g(x) = sin(x). Therefore, we can rewrite sin^3(x) as (sin(x))^3.
Next, we can differentiate the outer function f(u) = u^3. By using the power rule for differentiation, we obtain f'(u) = 3u^2.
Finally, we differentiate the inner function g(x) = sin(x) to get g'(x) = cos(x).
Now, applying the chain rule, we have:
d/dx [sin^3(x)] = 3(sin(x))^2 * cos(x)
Moving on to the next question:
To integrate sin^2(x) * cos(x) dx, we can use a substitution method. Let's make the substitution u = sin(x). Then, du/dx = cos(x), or du = cos(x) dx.
Now, we can rewrite the integral in terms of u:
∫ sin^2(x) * cos(x) dx = ∫ u^2 du
Integrating u^2 with respect to u gives us (1/3)u^3. Substituting back u = sin(x), we obtain:
∫ sin^2(x) * cos(x) dx = (1/3)sin^3(x) + C
where C is the constant of integration.
Lastly, let's find the equation of the tangent to the curve y = 2 sin(x - π/6) at the point where x = π/3.
To find the equation of the tangent, we need the slope and a point on the curve. The slope of the tangent is given by the derivative of the function at that point.
Taking the derivative of y = 2 sin(x - π/6), we have:
dy/dx = 2 cos(x - π/6)
Evaluating this derivative at x = π/3, we get:
dy/dx | x=π/3 = 2 cos(π/3 - π/6) = 2 cos(π/6) = √3
So, the slope of the tangent at x = π/3 is √3.
Now, let's find the corresponding y-value for x = π/3:
y = 2 sin(π/3 - π/6) = 2 sin(π/6) = 2 * (1/2) = 1
Therefore, the point on the curve is (π/3, 1).
Now, we can use the point-slope form of a line to write the equation of the tangent:
y - y1 = m(x - x1)
where (x1, y1) is the point (π/3, 1) and m is the slope √3:
y - 1 = √3(x - π/3)
Simplifying and rearranging:
y = √3x - √3π/3 + 1
Finally, this is the equation of the tangent to the curve y = 2 sin(x - π/6) at the point where x = π/3.