Q.Integrate the following function: $\int \frac{x^3 + 3x + 4}{\sqrt{x}} dx$

We rewrite the integrand by dividing each term by $\sqrt{x} = x^{1/2}$, converting the expression into a sum of power functions.

So the integral becomes:

Now apply the Power Rule for integration: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ for each term. …

The key to integrating a rational expression like this is to avoid trying to integrate the fraction as a whole. Instead, break it into separate fractions — one for each term in the numerator — and simplify each using the laws of exponents. Once every term is a simple power of $x$, the Power Rule for integration does all the work.

Why the Power Rule works here

The Power Rule says: for any real number $n \neq -1$,

Our integrand $\frac{x^3 + 3x + 4}{\sqrt{x}}$ is not yet in the form $x^n$, but we can rewrite it. Since $\sqrt{x} = x^{1/2}$, dividing each term in the numerator by $x^{1/2}$ gives a sum of terms like $x^{\text{something}}$. Then we apply the Power Rule to each term separately.

Step‑by‑step solution

1. Rewrite the square root as a power.

2. Split into separate fractions and simplify each using $x^a / x^b = x^{a-b}$.

Now the integrand is a clean sum of power functions.

3. Integrate term by term using the Power Rule.

• For $x^{5/2}$: $n = \frac{5}{2}$, so $n+1 = \frac{7}{2}$.

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

• For $4x^{-1/2}$: $n = -\frac{1}{2}$, so $n+1 = \frac{1}{2}$. …

Method: Dividing each term by a root denominator

For a numerator over $\sqrt x$, rewrite $\sqrt x=x^{1/2}$ and divide term by term into fractional powers.

Steps

Step 1: Split over $x^{1/2}$.

$\dfrac{x^3+3x+4}{x^{1/2}}=x^{5/2}+3x^{1/2}+4x^{-1/2}$, using $\dfrac{x^a}{x^{1/2}}=x^{a-1/2}$.

Step 2: Power-rule each fractional power. …

Common Mistakes

Mistake 1: Splitting the quotient into a quotient of integrals.

Why it's wrong: $\frac{\int(\cdots)}{\int\sqrt x}$ is not a valid operation. Correct approach: divide each numerator term by $x^{1/2}$ into a sum of powers, then integrate.

Mistake 2: Fumbling the reciprocal of a fractional power. …