Trigonometric Substitution Integration

Question

2.) ∫ ((9+25x^2)^(1/2)*dx)/(x^2)

Answer 1

x = (3/5)tan u

x^2 = (9/25)(tan u)^2

dx = (sec u)^2

integral [ (sec u)^2 * sqrt( 9 + 25 (9/25)(tan u)^2) / (tan u)^2]

integral [ (sec u)^2 * sqrt( 9 + 9 (tan u)^2) / (tan u)^2 ]

3 integral [ (sec u)^3 / (tan u)^2 ]

3 integral [ (csc u)^2 (sec u)]

3 integral [ (1 + (cot u)^2)(sec u) ]

3 integral [ sec u + cot u csc u ]

3 integral[sec u] + 3 integral [ cot u csc u]

integral of secant is ln| tan u + sec u| per the table

integral of [cot u csc u] = -csc u per the table