Final answer
To integrate x^n sin(x) dx, where n is a positive integer, we use integration by parts. Letting u = x^n and dv = sin(x) dx, we have du/dx = nx^(n-1) and v = -cos(x). Substituting these into the formula for integration by parts, we get:
∫x^n sin(x) dx = uv - \int vdu
Simplifying this, we get:
∫x^n sin(x) dx = -x^n cos(x) + n\int x^(n-1)cos(x) dx
This is our final answer, with the integral on the right-hand side still needing to be solved.
Step-by-step explanation
Integration by parts is a technique used to simplify integrals that cannot be solved using basic algebraic manipulations. It involves breaking down the integral into two parts, one of which can be easily integrated (in this case, the cosine function), and the other of which can be differentiated (in this case, the x^n function). By doing so, we can reduce the order of differentiation required for the second part, making it easier to integrate.
In our case, we let u = x^n and dv = sin(x) dx. This means that du/dx = nx^(n-1) and v = -cos(x). Substituting these into the formula for integration by parts, we get:
∫u dv = uv - \int vdu
Here, we have uv at the beginning of the integral, which is simply the product of u and v evaluated at the endpoints of the interval of integration. The second term in the equation represents an integral that involves v and du/dx. By integrating this term, we can simplify our original integral.
In our case, we have v = -cos(x), so du/dx = nx^(n-1). Substituting these into our formula for integration by parts, we get:
∫x^n sin(x) dx = -x^n cos(x) + n\int x^(n-1)cos(x) dx
The first term in this expression represents a simple product of functions that can be easily evaluated. The second term involves an integral that still needs to be solved. However, because we have reduced the order of differentiation required for this term (from n to n-1), it should now be easier to integrate than if we had left it as x^n sin(x). This is the power of integration by parts - it allows us to break down complex integrals into simpler parts that are easier to handle.