integrate x(xsin(x)dx from 0 to pi
To integrate the function x(x*sin(x)) from 0 to π, we can use the method of integration by parts.
Integration by parts is a technique based on the product rule for differentiation. The formula for integration by parts is:
∫ u dv = uv - ∫ v du
Let's assign u and dv as follows:
u = x, and dv = xsin(x)dx.
To determine du and v, we need to differentiate u and integrate dv:
du = dx (the derivative of x is 1),
v = ∫ dv = ∫ xsin(x)dx.
To integrate v, we can use integration by parts again. Let's assign u and dv for this step as follows:
u = x, and dv = sin(x)dx.
Now we can find du and v for this step:
du = dx (the derivative of x is 1),
v = ∫ dv = ∫ sin(x)dx = -cos(x).
Using the formula for integration by parts, we can rewrite the original integral as follows:
∫ x(x*sin(x))dx
= x * (-cos(x)) - ∫ (-cos(x)) dx.
We simplify the expression:
∫ x(x*sin(x))dx = -x*cos(x) + ∫ cos(x) dx.
Integrating cos(x) gives us:
∫ cos(x) dx = sin(x) + C,
where C is the constant of integration.
Now we can substitute the values of the limits of integration, 0 and π, into the expression:
∫[0 to π] x(x*sin(x))dx
= [ -x*cos(x) + sin(x) ] evaluated from 0 to π.
Evaluating at π:
[ -π*cos(π) + sin(π) ] = [ -π*(-1) + 0 ] = π,
and evaluating at 0:
[ -0*cos(0) + sin(0) ] = [ 0 + 0 ] = 0.
Therefore, the value of ∫[0 to π] x(x*sin(x))dx is π.