Question

Evaluate the integral. (Use C for the constant of
integration.)
(x2 + 8x) cos(x) dx

Answer 1

The integral of (x2 + 8x)cos(x)dx is -x2sin(x) - 4xcos(x) + C.

Explanation:

This integral can be solved using integration by parts. Integration by parts involves finding the integral of two functions multiplied together, and then taking the anti-derivative of the product of those two functions. In this case, the two functions that are being multiplied together are (x2 + 8x) and cos(x). To find the integral of these two functions, we can use the formula for integration by parts:

Integral of udv = uv - integral of vdu

We will let u = x2 + 8x and dv = cos(x)dx. This means that v = -sin(x) and du = (2x + 8)dx. Plugging these values into the integration by parts formula, we get:

Integral of (x2 + 8x)cos(x)dx = (x2 + 8x)(-sin(x)) - integral of (-sin(x))(2x + 8)dx

Integrating the second part of this equation, we get:

Integral of (x2 + 8x)cos(x)dx = (x2 + 8x)(-sin(x)) + x2cos(x) + 8cos(x) + C

Simplifying this equation, we get:

Integral of (x2 + 8x)cos(x)dx = -x2sin(x) - 4xcos(x) + C

Therefore, the integral of (x2 + 8x)cos(x)dx is -x2sin(x) - 4xcos(x) + C.