Final answer:
To evaluate the given integral using integration by parts, assign the values u = ln(2x^5) and dv = 5x^2 dx. Apply the integration by parts formula to find the result and simplify the expression.
Step-by-step explanation:
To evaluate the integral, we can use the integration by parts method. Let's start by assigning the values of u and dv. In this case, u = ln(2x^5) and dv = 5x^2 dx.
According to the integration by parts formula, we have:
∫ u dv = uv - ∫ v du,
Let's find du by differentiating u:
du = d(ln(2x^5)) = (1/(2x^5))(10x^4) = 5/x dx,
Now, we can substitute these values into the formula:
∫ 5ln(2x^5) x^2 dx = uv - ∫ v du = ln(2x^5) * x^2 - ∫ 5/x * x^2 dx.
Simplifying further, we have:
∫ 5ln(2x^5) x^2 dx = x^2 ln(2x^5) - 5 ∫ x dx = x^2 ln(2x^5) - (5/2)x^2 + C,
where C is the constant of integration.