find the following integral
(1) dx/25+(x-5)^2
(2) cos^8xsinxdx
(1) Do you mean dx/(25+(x-5)^2) or dx(1/25+(x-5)^2)?
(2) (cos(x))^8sin(x)dx
let u = cos(x)
du = - sin(x)
= integral(-u^8du)
= -(u^9)/9 +C
= -((cos(x))^9)/9 +C
To find the integral of function (1):
∫ [(1/25) + (x-5)^2] dx
You can break this integral down into two separate integrals, since the addition of terms allows you to integrate each term separately:
∫ (1/25) dx + ∫ (x-5)^2 dx
For the first term, ∫ (1/25) dx, 1/25 is a constant. The integral of a constant is simply the constant multiplied by x:
(1/25) ∫ dx = (1/25) * x = x/25
Now let's find the integral of the second term, ∫ (x-5)^2 dx. You can expand and simplify the expression:
∫ (x^2 - 10x + 25) dx
To integrate each term, use the power rule of integration:
∫ x^2 dx = (1/3) x^3
∫ -10x dx = -5x^2
∫ 25 dx = 25x
Adding up these integrals gives:
∫ (x-5)^2 dx = (1/3) x^3 - 5x^2 + 25x
Now, combine both integrals:
∫ [(1/25) + (x-5)^2] dx = ∫ (1/25) dx + ∫ (x-5)^2 dx
= x/25 + (1/3) x^3 - 5x^2 + 25x
That is the integral of the given function.
Now, let's move on to the integral of function (2):
∫ cos^8(x) sin(x) dx
To solve this integral, we can use integration by parts, which is a technique based on the product rule of differentiation. The formula for integration by parts is:
∫ u dv = u v - ∫ v du
Here, we can select u = cos^8(x) and dv = sin(x) dx. Hence, we differentiate u to obtain du and integrate dv to obtain v.
Let's begin by finding du:
du = d(cos^8(x))
= -8 cos^7(x) sin(x) dx
Next, find v by integrating dv:
v = ∫ sin(x) dx = -cos(x)
Now, we can apply the integration by parts formula:
∫ cos^8(x) sin(x) dx = cos^8(x) (-cos(x)) - ∫ [-cos(x)] (-8 cos^7(x) sin(x)) dx
= -cos^9(x) + 8 ∫ cos^7(x) sin^2(x) dx
Notice that in the second integral, we have sin^2(x). We can use the identity sin^2(x) = 1 - cos^2(x) to simplify further:
∫ cos^8(x) sin(x) dx = -cos^9(x) + 8 ∫ cos^7(x) (1 - cos^2(x)) dx
Now, let's expand the product:
∫ cos^8(x) sin(x) dx = -cos^9(x) + 8 ∫ cos^7(x) dx - 8 ∫ cos^9(x) dx
To integrate each term, you can use the power rule of integration. The final solution would depend on whether you want the definite or indefinite integral, and if a specific interval or range is provided.
I hope this explanation helps you understand how to approach these integrals!