To find the integral of the function sin(x) * x, we will use a method called integration by parts. This technique is particularly useful when dealing with the product of two functions, as is the case here.
The formula for integration by parts is:
∫u dv = uv – ∫v du
For our integral, we will choose:
- u = x ⇒ du = dx
- dv = sin(x) dx ⇒ v = -cos(x)
Now, we can apply the integration by parts formula:
∫sin(x) * x dx = uv – ∫v du
Substituting in our values:
∫sin(x) * x dx = -x * cos(x) – ∫(-cos(x)) dx
This simplifies to:
-x * cos(x) + ∫cos(x) dx
Now, we can integrate cos(x):
∫cos(x) dx = sin(x)
Putting it all together, we have:
∫sin(x) * x dx = -x * cos(x) + sin(x) + C
Where C is the constant of integration. Therefore, the integral of sin(x) * x is:
∫ sin(x) * x dx = -x * cos(x) + sin(x) + C