evaluate the integral. (use C for the constant of integration.)
integral of 9xcos(4x)dx
To evaluate the integral ∫ 9x cos(4x) dx, we will use integration by parts.
Let u = x and dv = 9cos(4x) dx.
Then, du = dx and v = ∫ 9cos(4x) dx.
Applying the integration rules for cos(mx), where m is a constant, we have:
v = ∫ 9cos(4x) dx = 9 * (1/4) * sin(4x) = (9/4) * sin(4x) + C1, where C1 is the constant of integration.
Now, using the integration by parts formula:
∫ u dv = uv - ∫ v du,
we get:
∫ 9x cos(4x) dx = x * [(9/4) * sin(4x)] - ∫ (9/4) * sin(4x) dx.
Simplifying this expression, we have:
∫ 9x cos(4x) dx = (9/4) * x * sin(4x) - (9/4) * ∫ sin(4x) dx.
Applying the integration rules for sin(mx), where m is a constant, we find:
∫ sin(4x) dx = -(1/4) * cos(4x) + C2, where C2 is a new constant of integration.
Substituting this back into our original expression, we have:
∫ 9x cos(4x) dx = (9/4) * x * sin(4x) - (9/4) * [-(1/4) * cos(4x) + C2].
Simplifying and combining terms, we get:
∫ 9x cos(4x) dx = (9/4) * x * sin(4x) + (9/16) * cos(4x) + C,
where C = (9/4) * C2 is the overall constant of integration.