Integrate ∫ xsinxcosx dx
no
I=∫ xsinxcosx dx (subst. double angle)
=(1/2)∫ xsin(2x) dx
Integrate by parts
I=-(1/4)xcos(2x)+(1/4)∫ cos(2x) dx
=-(1/4)xcos(2x)+(1/8)sin(2x)+C
To find the integral of ∫ xsin(x)cos(x) dx, we can use integration by parts. The formula for integration by parts is ∫ u dv = uv - ∫ v du.
Let's choose u = x and dv = sin(x)cos(x) dx.
To find du, we differentiate u with respect to x: du = dx.
To find v, we integrate dv. To integrate sin(x)cos(x) dx, we can use the substitution method.
Let's substitute u = sin(x) and differential du = cos(x) dx. Then dv = u' dx = cos(x) dx.
So, v = ∫ dv = ∫ cos(x) dx = sin(x).
Now we can use the integration by parts formula:
∫ xsin(x)cos(x) dx = uv - ∫ v du
∫ xsin(x)cos(x) dx = x(sin(x)) - ∫ sin(x) dx
∫ xsin(x)cos(x) dx = xsin(x) + ∫ sin(x) dx
∫ xsin(x)cos(x) dx = xsin(x) - cos(x) + C
Therefore, the antiderivative of ∫ xsin(x)cos(x) dx is xsin(x) - cos(x) + C, where C is the constant of integration.
To integrate ∫ xsin(x)cos(x) dx, we will use integration by parts. The formula for integration by parts is:
∫ u * v dx = u * ∫ v dx - ∫ u' * (∫ v dx) dx
Let's assign u and v:
u = x (differentiate to get u' = 1)
v = sin(x)cos(x) (integrate to get ∫ v dx)
We need to find ∫ v dx, so let's solve for this integral first.
To integrate ∫ sin(x)cos(x) dx, we will use a trigonometric identity. The identity we will use is:
sin(2x) = 2sin(x)cos(x)
Rearranging the formula, we get:
sin(x)cos(x) = 1/2 * sin(2x)
Now let's integrate ∫ sin(x)cos(x) dx:
∫ sin(x)cos(x) dx = 1/2 * ∫ sin(2x) dx
Applying the integral of sin(2x), we have:
∫ sin(2x) dx = -(1/2) * cos(2x) + C
where C is the constant of integration.
Now we substitute back into the formula for integration by parts:
∫ xsin(x)cos(x) dx = u * ∫ v dx - ∫ u' * (∫ v dx) dx
= x * [-(1/2) * cos(2x) + C] - ∫ 1 * [-(1/2) * cos(2x) + C] dx
= - (1/2) * x * cos(2x) + C1 + (1/4) * sin(2x) + C2x + C
Therefore, the final answer is:
∫ xsin(x)cos(x) dx = - (1/2) * x * cos(2x) + (1/4) * sin(2x) + C1 + C2x + C
where C1, C2 are constants of integration.