Evaluate the integral
Int 7x ln(5x) dx
it should help to know that integral (x lnx) = 1/4 x^2 (2lnx - 1)
To evaluate the integral ∫7x ln(5x) dx, we can use integration by parts.
Integration by parts is a technique that allows us to express the integral of a product of two functions in terms of simpler integrals. It is based on the product rule for differentiation.
Let's break down the integral using the formula for integration by parts:
∫u dv = uv - ∫v du
In this case, let's choose:
u = ln(5x) and dv = 7x dx
To find du, we differentiate u with respect to x:
du = (1/x) dx
To find v, we integrate dv with respect to x:
v = ∫7x dx = (7/2) x^2
Using the formula for integration by parts, we can express the integral as:
∫7x ln(5x) dx = (7/2) x^2 ln(5x) - ∫(7/2) x^2 (1/x) dx
Simplifying the expression:
∫7x ln(5x) dx = (7/2) x^2 ln(5x) - (7/2) ∫x dx
Evaluating the indefinite integral of x with respect to x:
∫x dx = (1/2) x^2
Inserting this result back into the expression, we have:
∫7x ln(5x) dx = (7/2) x^2 ln(5x) - (7/2)(1/2) x^2
Simplifying further:
∫7x ln(5x) dx = (7/2) x^2 ln(5x) - (7/4) x^2 + C
where C is the constant of integration. Therefore, this is the evaluation of the integral.