a.
$
\eqalign{
& \int {{1 \over {1 + \sqrt x }}} \,\,dx = \cr
& \int {{{2\sqrt x } \over {1 + \sqrt x }}} \,\,{1 \over {2\sqrt x }}dx = \cr
& \int {{{2\sqrt x } \over {1 + \sqrt x }}} \,\,d\left( {\sqrt x } \right) = \cr
& \int {{{2u} \over {1 + u}}} \,\,du = \cr
& \int {2 - {2 \over {u + 1}}\,\,du = } \cr
& 2u - 2\ln (u + 1) + C = \cr
& 2\sqrt x - 2\ln \left( {\sqrt x + 1} \right) + C \cr}
$
b.
$
\eqalign{
& \int {{{2x + 1} \over {\sqrt {x + 1} }}\,\,dx} = \cr
& \int {2\left( {2x + 1} \right) \cdot {1 \over {2\sqrt {x + 1} }}\,\,dx} = \cr
& \int {2\left( {2x + 1} \right) \cdot d\left( {\sqrt {x + 1} } \right)} = \cr
& \int {2\left( {2\left( {u^2 - 1} \right) + 1} \right) \cdot du} = \cr
& \int {4u^2 - 2 \cdot du} = \cr
& {4 \over 3}u^3 - 2u + C \cr
& {4 \over 3}\left( {\sqrt {x + 1} } \right)^3 - 2\sqrt {x + 1} + C \cr
& \sqrt {x + 1} \left( {{4 \over 3}\left( {\sqrt {x + 1} } \right)^2 - 2} \right) + C \cr
& \sqrt {x + 1} \left( {{4 \over 3}\left( {x + 1} \right) - 2} \right) + C \cr
& \sqrt {x + 1} \left( {{4 \over 3}x - {2 \over 3}} \right) + C \cr
& {2 \over 3}\sqrt {x + 1} \left( {2x - 1} \right) + C \cr}
$
c.
$
\eqalign{
& \int {{{\tan (\ln (\sqrt x ))} \over x}} \,dx = \cr
& \int {\tan \left( {{1 \over 2}\ln (x)} \right)} \cdot {1 \over x}\,dx = \cr
& \int {\tan \left( {{1 \over 2}\ln (x)} \right)} \,d\left( {\ln (x)} \right) = \cr
& \downarrow u = \ln (x) \cr
& \int {\tan \left( {{1 \over 2}u} \right)} \,du = \cr
& \int {{{\sin \left( {{1 \over 2}u} \right)} \over {\cos \left( {{1 \over 2}u} \right)}}} \,du = \cr
& \int {{{ - 2} \over {\cos \left( {{1 \over 2}u} \right)}}} \cdot - {1 \over 2}\sin \left( {{1 \over 2}u} \right)\,du = \cr
& \int {{{ - 2} \over {\cos \left( {{1 \over 2}u} \right)}}} \cdot \,d\left( {\cos \left( {{1 \over 2}u} \right)} \right) = \cr
& \downarrow v = \cos \left( {{1 \over 2}u} \right) \cr
& \int {{{ - 2} \over v}} \cdot \,dv = \cr
& - 2\ln (v) + C = \cr
& - 2\ln \left( {\cos \left( {{1 \over 2}u} \right)} \right) + C = \cr
& - 2\ln \left( {\cos \left( {{1 \over 2}\ln (x)} \right)} \right) + C \cr
& - 2\ln \left( {\cos \left( {\ln (\sqrt x )} \right)} \right) + C \cr}
$
