What is the integral of 20 * (sinx)^3 * (cosx)^2?
y' = 20 * (sinx)^3 * (cosx)^2
= 20 sinx(sinx)^2(cosx)^2
= 20sinx(1 - cos^2 x)cos^2 x
= 20sinx cos^2 x - 20sinx cos^4 x
y = -20/3(cos^3 x) + 20/3(cos^5 x) + C, where C is a constant
let u = sin^2 x
then du = 2 sin x cos x dx
let dv = -3 sin x cos^2 x dx
then v = cos^3 x
u dv = -3 sin^3 x cos^2 x dx
so
let u = -20/3 sin^2 x
then du = -40/3 sin x cos x dx
let dv = -(1/3)sin x cos^2 x dx
then v = cos^3 x
now by parts
int u dv = u v - int v du
int (20 sin^3 x cos^2 x dx) =
-(20/3)sin^2 x cos^3 x +(40/3)int(cos^4 x sin x dx)
well remember the first term at the end and the (40/3) and work on the integral
int cos^4 x sin x dx = -(1/5)cos^5 x
I think you can get it from there
To find the integral of 20 * (sinx)^3 * (cosx)^2, we can use the technique of integration by parts.
Integration by parts is a method that allows us to integrate the product of two functions. The formula for integration by parts is:
∫ u * dv = u * v - ∫ v * du
Here, u and dv represent two different functions, and du and v represent their respective differentials.
To apply integration by parts, we need to choose which part of the function will be our u and which part will be our dv.
Let's select u = (sinx)^3 and dv = 20 * (cosx)^2 dx.
Next, we calculate du and v:
- Differentiating u, we get du = 3 * (sinx)^2 * cosx dx.
- Integrating dv, we get v = ∫ 20 * (cosx)^2 dx.
To find the integral of v, we can use the double-angle identity for cos(2x), which states that cos(2x) = 2 * (cosx)^2 - 1. Rearranging the terms, we can express (cosx)^2 in terms of cos(2x):
(cosx)^2 = (1 + cos(2x)) / 2.
Using this identity, we have:
v = ∫ 20 * (cosx)^2 dx
= ∫ 20 * [(1 + cos(2x)) / 2] dx
= ∫ 10 * (1 + cos(2x)) dx
= 10 * [x + (1/2) * sin(2x)] + C,
where C represents the constant of integration.
Now, we have all the necessary components to calculate the integral of the original function:
∫ 20 * (sinx)^3 * (cosx)^2 dx = u * v - ∫ v * du
= (sinx)^3 * [10 * (x + (1/2) * sin(2x))] - ∫ [10 * (x + (1/2) * sin(2x))] * 3 * (sinx)^2 * cosx dx.
Simplifying the expression, we get:
∫ 20 * (sinx)^3 * (cosx)^2 dx = 10 * (sinx)^3 * (x + (1/2) * sin(2x)) - ∫ 30 * (sinx)^2 * (x + (1/2) * sin(2x)) * cosx dx.
We can now repeat the integration by parts process on the new integral, or apply other integration techniques if needed, until we reach a point where we can evaluate the integral.