Question

${\int }_0^1\frac{dx}{x+\sqrt{x}}$ equals

• A. $ln2$
• B. $2\ln 2-2$
• C. $2ln2$
• D. $2ln2-1$

Answer 1

The correct option is C $2ln2$

Consider the given integral.

$I={\int }_0^1\frac{dx}{x+\sqrt{x}}$

$I={\int }_0^1\frac{dx}{\sqrt{x}(1+\sqrt{x})}$

Let $t=1+\sqrt{x}$

$\frac{dt}{dx}=\frac{1}{2\sqrt{x}}$

$2dt=\frac{1}{\sqrt{x}}dx$

Therefore,

$I=2{\int }_1^2\frac{dt}{t}$

$I=2{\left[\ln \left(t\right)\right]}_1^2$

$I=2[\ln \left(2\right)-\ln \left(1\right)]$

$I=2\ln \left(2\right)$

Hence, the value of integral is $2\ln \left(2\right)$.