Q.Integrate the following function: $\int \sqrt{x}(3x^2 + 2x + 3) dx$

The key idea is to rewrite the square root as a power and then apply the Power Rule for integration term by term.

First, rewrite $\sqrt{x}$ as $x^{1/2}$ and multiply through:

Now integrate each term using $\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$:

Simplify the coefficients: …

Before we jump into algebra, let’s see why this works. The Power Rule for integration is the reverse of the Power Rule for differentiation: if you know how to differentiate $x^n$, you already know how to integrate it. The only catch is that the exponent $n$ can be any real number — fractions, negatives, anything — as long as $n \neq -1$. Here, $\sqrt{x}$ is $x^{1/2}$, and the polynomial inside is a sum of powers. Multiplying them gives a sum of terms like $x^{1/2} \cdot x^2 = x^{5/2}$, which is still a power function. So the whole problem reduces to integrating a sum of power functions.

Now let’s work through it step by step.

1. Rewrite the square root as a power.

   $\sqrt{x} = x^{1/2}$. This lets us use the Power Rule cleanly.

2. Distribute $\sqrt{x}$ across the polynomial.

For each term, add the exponents: $x^{1/2} \cdot x^2 = x^{1/2 + 2} = x^{5/2}$, and similarly for the others.

So we get:

3. Integrate each term using the Power Rule. The rule: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$, provided $n \neq -1$.
   • For $3x^{5/2}$: $n = \frac{5}{2}$, so $n+1 = \frac{7}{2}$.

• For $2x^{3/2}$: $n = \frac{3}{2}$, $n+1 = \frac{5}{2}$.

• For $3x^{1/2}$: $n = \frac{1}{2}$, $n+1 = \frac{3}{2}$. $\int 3x^{1/2} dx = 3 \cdot \frac{x^{3/2}}{3/2} = 3 \cdot \frac{2}{3} x^{3/2} = 2 x^{3/2}$ …

Method: Distribute the root across a polynomial, then power-rule

This is the go-to method when a root of $x$ multiplies a polynomial, e.g. $\sqrt{x}\,(3x^2+2x+3)$: there is no product rule for integration, so you must expand into standard powers first.

Steps

Step 1: Convert the root to a fractional power.

Write $\sqrt{x}=x^{1/2}$ so the exponent law can be applied.

Step 2: Multiply the power into each term of the polynomial.

Using $x^{1/2}\cdot x^k = x^{k+1/2}$, every term becomes a single power of $x$. Keep the numerical coefficients out front. …

Common Mistakes

Mistake 1: Adding exponents incorrectly when distributing $\sqrt{x}$.

Why it's wrong: $x^{1/2}\cdot x^2=x^{5/2}$, not $x^{1}$ or $x^{2.5}$ written as $x^{3/2}$ — you add $\tfrac12+2=\tfrac52$. Correct approach: use $x^a x^b=x^{a+b}$ carefully on every term.

Mistake 2: Mishandling the fraction when dividing by $n+1$. …