Question

can you solve this problem​

Answer 1

$\rm\bf\dag\large\underline{Soluation: }$

$\rm\rightarrow\huge\int \frac{1 +x + \: \sqrt{x \: (1+ x)} }{ \sqrt{x} + \sqrt{1 + x} } \: dx$

$\rm \rightarrow\huge\int \frac{ \sqrt{1 + x} \: + \: \sqrt{ 1+ x \: } + \sqrt{x} }{ \sqrt{1 + x} \: + \sqrt{x} } \: dx$

$\rm \rightarrow\huge \int \: \sqrt{1 + x} \: \: \: \: \: dx$

$\rm \rightarrow\huge\frac{(x + 1) ^{ \frac{3}{2} } } { \: \frac{3}{2} } + c$

$\rm \rightarrow\huge\frac{2}{3}(1 + x)^{ \frac{3}{2} } \: + c$

$\rm\bf Your \: answer \: is, \\ \rm \frac{2}{3} (1 + x) ^{ \frac{3}{2} } \: + c$

Answer 2

Answer:

\rm\bf\dag\large\underline{Soluation: }†

Soluation:

\rm\rightarrow\huge\int \frac{1 +x + \: \sqrt{x \: (1+ x)} }{ \sqrt{x} + \sqrt{1 + x} } \: dx→∫

x

+

1+x

1+x+

x(1+x)

dx

\rm \rightarrow\huge\int \frac{ \sqrt{1 + x} \: + \: \sqrt{ 1+ x \: } + \sqrt{x} }{ \sqrt{1 + x} \: + \sqrt{x} } \: dx→∫

1+x

+

x

1+x

+

1+x

+

x

dx

\rm \rightarrow\huge \int \: \sqrt{1 + x} \: \: \: \: \: dx→∫

1+x

dx

\rm \rightarrow\huge\frac{(x + 1) ^{ \frac{3}{2} } } { \: \frac{3}{2} } + c→

2

3

(x+1)

2

3

+c

\rm \rightarrow\huge\frac{2}{3}(1 + x)^{ \frac{3}{2} } \: + c→

3

2

(1+x)

2

3

+c

\begin{gathered} \rm\bf Your \: answer \: is, \\ \rm \frac{2}{3} (1 + x) ^{ \frac{3}{2} } \: + c\end{gathered}

Youransweris,

3

2

(1+x)

2

3

+c