Q.Integrate the following function: The anti derivative of $\left(\sqrt{x} + \frac{1}{\sqrt{x}}\right)$ equals (A) $\frac{1}{3}x^{\frac{1}{3}} + 2x^{\frac{1}{2}} + C$ (B) $\frac{2}{3}x^{\frac{2}{3}} + \frac{1}{2}x^2 + C$ (C) $\frac{2}{3}x^{\frac{3}{2}} + 2x^{\frac{1}{2}} + C$ (D) $\frac{3}{2}x^{\frac{2}{3}} + \frac{1}{2}x^2 + C$

Concept: Power Rule Integration — $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$.

Rewrite each term in power form:

Integrate term by term:

Add the constant of integration: …

The core idea here is the Power Rule for Integration: for any real number $n \neq -1$,

This rule works because differentiation of $x^{n+1}$ brings down the exponent $n+1$, and dividing by it reverses that step. The only exception is $n = -1$, which gives the natural logarithm — but that doesn't appear here.

Our integrand is $\sqrt{x} + \frac{1}{\sqrt{x}}$. Before we can apply the Power Rule, we need to write each term as a power of $x$:

• $\sqrt{x} = x^{1/2}$
• $\frac{1}{\sqrt{x}} = x^{-1/2}$

So the problem becomes: find $\int \left( x^{1/2} + x^{-1/2} \right) dx$.

Now we integrate term by term.

1. Integrate $x^{1/2}$: Using the Power Rule with $n = \frac{1}{2}$:

Why $\frac{2}{3}$? Dividing by $\frac{3}{2}$ is the same as multiplying by $\frac{2}{3}$ — a common point of confusion.

2. Integrate $x^{-1/2}$: Here $n = -\frac{1}{2}$. Then $n+1 = \frac{1}{2}$, so:

Dividing by $\frac{1}{2}$ gives multiplication by $2$.

3. Combine the results: Adding the two antiderivatives and merging the constants into a single $C$: $\int \left( \sqrt{x} + \frac{1}{\sqrt{x}} \right) dx = \frac{2}{3} x^{3/2} + 2 x^{1/2} + C.$ …

Method: Evaluate an antiderivative MCQ by the power rule

For a "which option is the antiderivative" question, don't try to match options by eye — integrate the given function yourself, then compare.

Steps

Step 1: Rewrite roots as fractional powers.

Write $\sqrt{x}=x^{1/2}$ and $\dfrac{1}{\sqrt{x}}=x^{-1/2}$ so the power rule applies to each term.

Step 2: Apply the power rule term by term.

$\int x^n\,dx=\frac{x^{n+1}}{n+1}+C,\qquad n\neq-1.$ …

Common Mistakes

Mistake 1: Choosing an option that "looks like" differentiation.

Why it's wrong: distractors such as $\tfrac23 x^{2/3}$ come from dropping the exponent instead of raising it. Correct approach: for $\sqrt{x}=x^{1/2}$, integration gives exponent $3/2$, so $\tfrac23 x^{3/2}$.

Mistake 2: Mishandling $\dfrac{1}{\sqrt{x}}=x^{-1/2}$. …