How to Integrate 1/x+x sqrt(x)
Published On :2021-01-28 14:07:00
begin{array}{l} smallint frac{{bf{1}}}{{{bf{x}} + {bf{x}}sqrt {bf{x}} }}{rm{d}}{bf{x}}\\ By;Substitution\ ;;;;;;;;;;Suppose Rightarrow {z^2} = x;\ 2zdz = dx\ smallint frac{{2;{rm{z}}}}{{{z^2} + {z^3}}}{rm{d}}x\ smallint frac{{2;}}{{z({z^;} + 1)}}{rm{d}}x\ By;Using;Partial;Fraction\ smallint frac{2}{{z({z^;} + 1)}}{rm{d}}x = frac{A}{z} + frac{B}{{z + 1}};;\ ;;;;;;;;;Let Rightarrow z = 0 Rightarrow A = 2;;\ ;;;;;;;;;;;;;;;;Let Rightarrow z = - 1 Rightarrow B = - 2\ smallint frac{2}{z} - frac{2}{{z + 1}}{rm{d}}x\ 2ln left| z right| - 2ln left| {z + 1} right| + c\ Rightarrow 2ln left| {sqrt x } right| - 2ln left| {sqrt x + 1} right| + c end{array}