Integration Methods

Question

1. ∫⁵ 1 / x(6+5x) dx

¹

Examine the form of integral which involve a+bu. Let u=x, a=6, and b=5.

Answer 1

I am not sure about how to exploit the a + bu idea. However, the standard approach with this type of integrand is the method of partial fractions.

1/(x (6+ 5x) can be decomposed as (1/6) /x - (5/6) /(5 x + 6)

Since both x and 5 x + 6 are positive on the interval [1,5] the anti derivative is

(1/6) ln(x) - (1/6) ln(5 x +6) = (1/6) ln( x /( 5 x + 6))

So the definite integral is (1/6) [ ln(5/31) - ln( 1/11) ] = 0.09556