Question

Q. 19 plz help I want answer ples

Answer 1

We're given to evaluate,

$\longrightarrow I = \displaystyle\int\dfrac{\sqrt{x}+1}{x+\sqrt{x}}\:\:dx$

Taking $\sqrt{x}$ as common in the denominator,

$\longrightarrow I=\displaystyle\int\dfrac{\sqrt{x}+1}{\sqrt{x}(\sqrt{x}+1)}\:\:dx$

$\longrightarrow I=\displaystyle\int\dfrac{1}{\sqrt{x}}\:\:dx$

$\longrightarrow I = \displaystyle\int\dfrac{1}{x^{1/2}}\:\:dx$

$\longrightarrow I = \displaystyle\int x^{-1/2}\:\:dx$

$\longrightarrow I = \dfrac{x^{-1/2\:+\:1}}{ \quad \bigg(\dfrac{-1}{2}+1 \bigg) \quad } + C$

$\longrightarrow I = \dfrac{x^{1/2}}{\quad\bigg(\dfrac{1}{2}\bigg)\quad} + C$

$\longrightarrow \underline{\underline{I = 2\sqrt{x} + C}}$