Integration helpppppp

Question

Integrate  ∫ (8x³+20x²+51x+124)/(4x²+25) dx with arctan and ln in the answer... I think that the first step is to take it and divide it. This gave me a 2x+5 and a remainder of x-1. I think that's correct... I am stuck though. I think you set it up with the numerator = to A/ quotient + B/ remainder...

Answer 1

You are correct in your division. You put the remainder over the divisor so

(8x³+20x²+51x+124)/(4x²+25) = 2x + 5 + (x - 1)/(4x²+25)

Now divide the last term up into two parts to make the integrand:

2x + 5 + x/(4x²+25) - 1/(4x²+25)

So the integral becomes:

∫2x dx + ∫5 dx + ∫x/(4x²+25) dx - 1/(4x²+25) dx

The antiderivative of the first two terms are "x² + 5x"

To find the antiderivative of the 3rd term use a u-sub where u = 4x² + 25 and you will find the antiderivative is (1/8)ln(4x²+25)

To find the antiderivative of the 4th term, do a u-sub where u = (2/5)x and you'll find the antiderivative is (1/10)arctan((2/5)x)