Sure, I'd be happy to help with that. Let's go through each of the integrals one by one.
a) ∫(x+5)(x−3)dx
This integral can be calculated by first expanding the given integral to x^2 + 2x - 15. Then, integrate each term individually, which gives us (1/3)x^3 + x^2 - 15x.
b) ∫ (x+1) / (x^2 - 1) dx
This integral looks complicated, but it's very straightforward once you recognize it as a standard form. The indefinite integral of (x+1)/(x^2 - 1) can be directly solved using the logarithm integral rule. The result is log |x - 1|.
c) ∫ (1 / x) * (1 - x * e^x) dx
Split the integral into two separate integrals: ∫(1/x)dx - ∫(x*e^x)dx. Now, ∫(1/x)dx gives us log|x|, and ∫(x*e^x)dx can be solved using the formula for integration by parts, which is ∫udv=uv-∫vdu. Here, we let u=x and dv=e^x dx. The result is log|x| - e^x.
d) ∫ (6/x + 4/(x^3)) ^ (1/3) dx
This is a complex integral and does not have a standard integral form. The result of this integral can be represented as 1.25992104989487 * ∫ (3/x + 2/x^3)^(1/3) dx. The constant 1.25992104989487 is the result of the cube root of 2 (2^(1/3)) when evaluated numerically. The remaining integral is left as it is because it doesn't have a standard integral result that can be represented in terms of elementary functions.