Homework Sourseevaluate the improper integralYou must show work If the inte
evaluate the improper integral.You must show work. If the integral does not converge, state that the integral is divergent. definite integral 1 to infinite (5/(8x(x+1)^2) dx
A) -0.746
B) -5.965
C)0.625
D) 0.120
A) -0.746
B) -5.965
C)0.625
D) 0.120
Solution
? 5 / (8x(x+1)²) dx = 5/8 ? 1/(x(x+1)²) dx = 5/8 ? (1/x - 1/(x+1) - 1/(x+1)²) dx . . . . using partial fraction decomposition = 5/8 (ln(x) - ln(x+1) + 1/(x+1)) = 5/8 (ln(x/(x+1)) + 1/(x+1)) ______________________________ ?[from 1 to 8] 5/(8x(x+1)²) dx = lim[n?8] ?1n 5/(8x(x+1)²) dx = lim[n?8] 5/8 (ln(x/(x+1)) + 1/(x+1)) |1n = lim[n?8] 5/8 (ln(n/(n+1)) + 1/(n+1)) - 5/8 (ln(x/(x+1)) + 1/(x+1)) = 5/8 (ln(1) + 0) - 5/8 (ln(1/2) + 1/2) = 0 - 5/8 (ln(1/2) + 1/2) = 5/8 (-ln(1/2) - 1/2) = 5/8 (ln(2) - 1/2) ˜ 0.12071
