Correct Answer is : 2log⁡(1+x)+c 2 (1+{x})+c2log(1+x)+c, where c is the constant of integration.

{x}+c, where c is the constant of integration.

({x}+x)+c, where c is the constant of integration.

(1+{x})+c, where c is the constant of integration.

2 (1+{x})+c, where c is the constant of integration.

Correct Answer is : 2log⁡(1+x)+c 2 (1+{x})+c2log(1+x)+c, where c is the constant of integration.

Test Index

MHTCET 23 Apr 2025 Shift 1 Paper

Question : 123 of 150

Marks: +1, -0

\int \frac{d x}{\sqrt{x}+x}=

Solution:

Concept:

The integrand is simplified by factoring

\sqrt{x}

and using a substitution to reduce it to a standard logarithmic integral.

Explanation:

Given integral is

I = \int \frac{dx}{\sqrt{x}+x}.

Factor the denominator as

\sqrt{x}+x = \sqrt{x}(1+\sqrt{x}).

So,

I = \int \frac{dx}{\sqrt{x}(1+\sqrt{x})}.

Let

t = 1+\sqrt{x}

Then

\frac{dt}{dx} = \frac{1}{2\sqrt{x}} \Rightarrow \frac{dx}{\sqrt{x}} = 2\,dt.

Substituting into the integral,

I = \int \frac{2\,dt}{t} = 2\int \frac{dt}{t}.

This gives

I = 2\log|t| + c.

Substitute back

t = 1+\sqrt{x}

I = 2\log(1+\sqrt{x}) + c.

Answer:

2\log(1+\sqrt{x}) + c

Hence, the correct option is D.

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